Editing Banach space (section)

==General theory==

===Linear operators, isomorphisms===
<!-- This section is linked from [[Operator]] -->
{{main|Bounded operator}}
If <math>X</math> and <math>Y</math> are normed spaces over the same [[ground field]] <math>\mathbb{K},</math> the set of all [[Continuous function (topology)|continuous]] [[Linear transformation|<math>\mathbb{K}</math>-linear maps]] <math>T : X \to Y</math> is denoted by <math>B(X, Y).</math> In infinite-dimensional spaces, not all linear maps are continuous. A linear mapping from a normed space <math>X</math> to another normed space is continuous if and only if it is [[bounded operator|bounded]] on the closed [[Unit sphere|unit ball]] of <math>X.</math> Thus, the vector space <math>B(X, Y)</math> can be given the [[operator norm]]
<math display=block>\|T\| = \sup \{\|Tx\|_Y \mid x\in X,\ \|x\|_X \leq 1\}.</math>

For <math>Y</math> a Banach space, the space <math>B(X, Y)</math> is a Banach space with respect to this norm.  In categorical contexts, it is sometimes convenient to restrict the [[Hom space|function space]] between two Banach spaces to only the [[short map]]s; in that case the space <math>B(X,Y)</math> reappears as a natural [[bifunctor]].<ref name=Ban1Cat>{{cite web|website=Annoying Precision|title=Banach spaces (and Lawvere metrics, and closed categories)|date=June 23, 2012|author=Qiaochu Yuan|url=https://qchu.wordpress.com/2012/06/23/banach-spaces-and-lawvere-metrics-and-closed-categories/}}</ref>

If <math>X</math> is a Banach space, the space <math>B(X) = B(X, X)</math> forms a unital [[Banach algebra]]; the multiplication operation is given by the composition of linear maps.

If <math>X</math> and <math>Y</math> are normed spaces, they are '''isomorphic normed spaces''' if there exists a linear bijection <math>T : X \to Y</math> such that <math>T</math> and its inverse <math>T^{-1}</math> are continuous. If one of the two spaces <math>X</math> or <math>Y</math> is complete (or [[Reflexive space|reflexive]], [[Separable space|separable]], etc.) then so is the other space. Two normed spaces <math>X</math> and <math>Y</math> are ''isometrically isomorphic'' if in addition, <math>T</math> is an [[isometry]], that is, <math>\|T(x)\| = \|x\|</math> for every <math>x</math> in <math>X.</math> The [[Banach–Mazur distance]] <math>d(X, Y)</math> between two isomorphic but not isometric spaces <math>X</math> and <math>Y</math> gives a measure of how much the two spaces <math>X</math> and <math>Y</math> differ.

====Continuous and bounded linear functions and seminorms====
Every [[continuous linear operator]] is a [[bounded linear operator]] and if dealing only with normed spaces then the converse is also true. That is, a [[linear operator]] between two normed spaces is [[Bounded linear operator|bounded]] if and only if it is a [[continuous function]]. So in particular, because the scalar field (which is <math>\R</math> or <math>\Complex</math>) is a normed space, a [[linear functional]] on a normed space is a [[bounded linear functional]] if and only if it is a [[continuous linear functional]]. This allows for continuity-related results (like those below) to be applied to Banach spaces. Although boundedness is the same as continuity for linear maps between normed spaces, the term "bounded" is more commonly used when dealing primarily with Banach spaces.

If <math>f : X \to \R</math> is a [[subadditive function]] (such as a norm, a [[sublinear function]], or real linear functional), then{{sfn|Narici|Beckenstein|2011|pp=192-193}} <math>f</math> is [[Continuity at a point|continuous at the origin]] if and only if <math>f</math> is [[uniformly continuous]] on all of <math>X</math>; and if in addition <math>f(0) = 0</math> then <math>f</math> is continuous if and only if its [[absolute value]] <math>|f| : X \to [0, \infty)</math> is continuous, which happens if and only if <math>\{x \in X \mid |f(x)| < 1\}</math> is an open subset of <math>X.</math>{{sfn|Narici|Beckenstein|2011|pp=192-193}}<ref group=note>The fact that <math>\{x \in X \mid |f(x)| < 1\}</math> being open implies that <math>f : X \to \R</math> is continuous simplifies proving continuity because this means that it suffices to show that <math>\{x \in X \mid |f(x) - f(x_0)| < r\}</math> is open for <math>r := 1</math> and at <math>x_0 := 0</math> (where <math>f(0) = 0</math>) rather than showing this for {{em|all}} real <math>r > 0</math> and {{em|all}} <math>x_0 \in X.</math></ref> 
And very importantly for applying the [[Hahn–Banach theorem]], a linear functional <math>f</math> is continuous if and only if this is true of its [[real part]] <math>\operatorname{Re} f</math> and moreover, <math>\|\operatorname{Re} f\| = \|f\|</math> and [[Real and imaginary parts of a linear functional|the real part <math>\operatorname{Re} f</math> completely determines]] <math>f,</math> which is why the Hahn–Banach theorem is often stated only for real linear functionals.
Also, a linear functional <math>f</math> on <math>X</math> is continuous if and only if the [[seminorm]] <math>|f|</math> is continuous, which happens if and only if there exists a continuous seminorm <math>p : X \to \R</math> such that <math>|f| \leq p</math>; this last statement involving the linear functional <math>f</math> and seminorm <math>p</math> is encountered in many versions of the Hahn–Banach theorem.

===Basic notions===

The Cartesian product <math>X \times Y</math> of two normed spaces is not canonically equipped with a norm. However, several equivalent norms are commonly used,<ref>{{harvtxt|Banach|1932|p=182}}</ref> such as
<math display=block>\|(x, y)\|_1 = \|x\| + \|y\|, \qquad \|(x, y)\|_\infty = \max(\|x\|, \|y\|)</math>
which correspond (respectively) to the [[coproduct]] and [[product (category theory)|product]] in the category of Banach spaces and short maps (discussed above).<ref name=Ban1Cat />  For finite (co)products, these norms give rise to isomorphic normed spaces, and the product <math>X \times Y</math> (or the direct sum <math>X \oplus Y</math>) is complete if and only if the two factors are complete.

If <math>M</math> is a [[Closed set|closed]] [[linear subspace]] of a normed space <math>X,</math> there is a natural norm on the quotient space <math>X / M,</math>
<math display=block>\|x + M\| = \inf\limits_{m \in M} \|x + m\|.</math>

The quotient <math>X / M</math> is a Banach space when <math>X</math> is complete.<ref name="Caro17">see pp.&nbsp;17–19 in {{harvtxt|Carothers|2005}}.</ref> The quotient map from <math>X</math> onto <math>X / M,</math> sending <math>x \in X</math> to its class <math>x + M,</math> is linear, onto, and of norm <math>1,</math> except when <math>M = X,</math> in which case the quotient is the null space.

The closed linear subspace <math>M</math> of <math>X</math> is said to be a ''[[complemented subspace]]'' of <math>X</math> if <math>M</math> is the [[Range of a function|range]] of a [[Surjection|surjective]] bounded linear [[Projection (linear algebra)|projection]] <math>P : X \to M.</math> In this case, the space <math>X</math> is isomorphic to the direct sum of <math>M</math> and <math>\ker P,</math> the kernel of the projection <math>P.</math>

Suppose that <math>X</math> and <math>Y</math> are Banach spaces and that <math>T \in B(X, Y).</math> There exists a canonical factorization of <math>T</math> as<ref name="Caro17" />
<math display=block>T = T_1 \circ \pi, \quad T : X \overset{\pi}{{}\longrightarrow{}} X/\ker T \overset{T_1}{{}\longrightarrow{}} Y</math>
where the first map <math>\pi</math> is the quotient map, and the second map <math>T_1</math> sends every class <math>x + \ker T</math> in the quotient to the image <math>T(x)</math> in <math>Y.</math> This is well defined because all elements in the same class have the same image. The mapping <math>T_1</math> is a linear bijection from <math>X/\ker T</math> onto the range <math>T(X),</math> whose inverse need not be bounded.

===Classical spaces===

Basic examples<ref>see {{harvtxt|Banach|1932}}, pp.&nbsp;11-12.</ref> of Banach spaces include: the [[Lp space]]s <math>L^p</math> and their special cases, the [[sequence space (mathematics)|sequence spaces]] <math>\ell^p</math> that consist of scalar sequences indexed by [[natural number]]s <math>\N</math>; among them, the space <math>\ell^1</math> of [[Absolute convergence|absolutely summable]] sequences and the space <math>\ell^2</math> of square summable sequences; the space <math>c_0</math> of sequences tending to zero and the space <math>\ell^{\infty}</math> of bounded sequences; the space <math>C(K)</math> of continuous scalar functions on a compact Hausdorff space <math>K,</math> equipped with the max norm,
<math display=block>\|f\|_{C(K)} = \max \{ |f(x)| \mid x \in K \}, \quad f \in C(K).</math>

According to the [[Banach–Mazur theorem]], every Banach space is isometrically isomorphic to a subspace of some <math>C(K).</math><ref>see {{harvtxt|Banach|1932}}, Th.&nbsp;9 p.&nbsp;185.</ref> For every separable Banach space <math>X,</math> there is a closed subspace <math>M</math> of <math>\ell^1</math> such that <math>X := \ell^1 / M.</math><ref>see Theorem&nbsp;6.1, p.&nbsp;55 in {{harvtxt|Carothers|2005}}</ref>

Any [[Hilbert space]] serves as an example of a Banach space. A Hilbert space <math>H</math> on <math>\mathbb{K} = \Reals, \Complex</math> is complete for a norm of the form
<math display=block>\|x\|_H = \sqrt{\langle x, x \rangle},</math>
where
<math display=block>\langle \cdot, \cdot \rangle : H \times H \to \mathbb{K}</math>
is the [[Inner product space|inner product]], linear in its first argument that satisfies the following:
<math display=block>\begin{align}
\langle y, x \rangle &= \overline{\langle x, y \rangle}, \quad \text{ for all } x, y \in H \\
\langle x, x \rangle & \geq 0, \quad \text{ for all } x \in H \\
\langle x,x \rangle = 0 \text{ if and only if } x &= 0.
\end{align}</math>

For example, the space <math>L^2</math> is a Hilbert space.

The [[Hardy space]]s, the [[Sobolev space]]s are examples of Banach spaces that are related to <math>L^p</math> spaces and have additional structure. They are important in different branches of analysis, [[Harmonic analysis]] and [[Partial differential equation]]s among others.

===Banach algebras===

A ''[[Banach algebra]]'' is a Banach space <math>A</math> over <math>\mathbb{K} = \R</math> or <math>\Complex,</math> together with a structure of [[Algebra over a field|algebra over <math>\mathbb{K}</math>]], such that the product map <math>A \times A \ni (a, b) \mapsto ab \in A</math> is continuous. An equivalent norm on <math>A</math> can be found so that <math>\|ab\| \leq \|a\| \|b\|</math> for all <math>a, b \in A.</math>

====Examples====

* The Banach space <math>C(K)</math> with the pointwise product, is a Banach algebra.
* The [[disk algebra]] <math>A(\mathbf{D})</math> consists of functions [[Holomorphic function|holomorphic]] in the open unit disk <math>D \subseteq \Complex</math> and continuous on its [[Closure (topology)|closure]]: <math>\overline{\mathbf{D}}.</math> Equipped with the max norm on <math>\overline{\mathbf{D}},</math> the disk algebra <math>A(\mathbf{D})</math> is a closed subalgebra of <math>C\left(\overline{\mathbf{D}}\right).</math>
* The [[Wiener algebra]] <math>A(\mathbf{T})</math> is the algebra of functions on the unit circle <math>\mathbf{T}</math> with absolutely convergent Fourier series. Via the map associating a function on <math>\mathbf{T}</math> to the sequence of its Fourier coefficients, this algebra is isomorphic to the Banach algebra <math>\ell^1(Z),</math> where the product is the [[Convolution#Discrete convolution|convolution]] of sequences.
* For every Banach space <math>X,</math> the space <math>B(X)</math> of bounded linear operators on <math>X,</math> with the composition of maps as product, is a Banach algebra.
* A [[C*-algebra]] is a complex Banach algebra <math>A</math> with an [[Antilinear map|antilinear]] [[Involution (mathematics)|involution]] <math>a \mapsto a^*</math> such that <math>\|a^* a\| = \|a\|^2.</math> The space <math>B(H)</math> of bounded linear operators on a Hilbert space <math>H</math> is a fundamental example of C*-algebra. The [[Gelfand–Naimark theorem]] states that every C*-algebra is isometrically isomorphic to a C*-subalgebra of some <math>B(H).</math> The space <math>C(K)</math> of complex continuous functions on a compact Hausdorff space <math>K</math> is an example of commutative C*-algebra, where the involution associates to every function <math>f</math> its [[complex conjugate]] <math>\overline{f}.</math>

===Dual space===
{{main|Dual space}}
If <math>X</math> is a normed space and <math>\mathbb{K}</math> the underlying [[Field (mathematics)|field]] (either the [[Real number|real]]s or the [[complex number]]s), the ''[[continuous dual space]]'' is the space of continuous linear maps from <math>X</math> into <math>\mathbb{K},</math> or ''continuous linear functionals''.
The notation for the continuous dual is <math>X' = B(X, \mathbb{K})</math> in this article.<ref>Several books about functional analysis use the notation <math>X^*</math> for the continuous dual, for example {{harvtxt|Carothers|2005}}, {{harvtxt|Lindenstrauss|Tzafriri|1977}}, {{harvtxt|Megginson|1998}}, {{harvtxt|Ryan|2002}}, {{harvtxt|Wojtaszczyk|1991}}.</ref> 
Since <math>\mathbb{K}</math> is a Banach space (using the [[absolute value]] as norm), the dual <math>X'</math> is a Banach space, for every normed space <math>X.</math> The [[Dixmier–Ng theorem]] characterizes the dual spaces of Banach spaces.

The main tool for proving the existence of continuous linear functionals is the [[Hahn–Banach theorem]].

{{math theorem|name=Hahn–Banach theorem|math_statement=Let <math>X</math> be a [[vector space]] over the field <math>\mathbb{K} = \R, \Complex.</math> Let further
* <math>Y \subseteq X</math> be a [[linear subspace]],
* <math>p : X \to \R</math> be a [[sublinear function]] and
* <math>f : Y \to \mathbb{K}</math> be a [[linear functional]] so that <math>\operatorname{Re}(f(y)) \leq p(y)</math> for all <math>y \in Y.</math> 
Then, there exists a linear functional <math>F : X \to \mathbb{K}</math> so that <math display=block>F\big\vert_Y = f, \quad \text{ and } \quad \text{ for all } x \in X, \ \ \operatorname{Re}(F(x)) \leq p(x).</math>}}

In particular, every continuous linear functional on a subspace of a normed space can be continuously extended to the whole space, without increasing the norm of the functional.<ref>Theorem&nbsp;1.9.6, p.&nbsp;75 in {{harvtxt|Megginson|1998}}</ref> 
An important special case is the following: for every vector <math>x</math> in a normed space <math>X,</math> there exists a continuous linear functional <math>f</math> on <math>X</math> such that
<math display=block>f(x) = \|x\|_X, \quad \|f\|_{X'} \leq 1.</math>

When <math>x</math> is not equal to the <math>\mathbf{0}</math> vector, the functional <math>f</math> must have norm one, and is called a ''norming functional'' for <math>x.</math>

The [[Hahn–Banach separation theorem]] states that two disjoint non-empty [[convex set]]s in a real Banach space, one of them open, can be separated by a closed [[Affine space|affine]] [[hyperplane]]. 
The open convex set lies strictly on one side of the hyperplane, the second convex set lies on the other side but may touch the hyperplane.<ref>see also Theorem&nbsp;2.2.26, p.&nbsp;179 in {{harvtxt|Megginson|1998}}</ref>

A subset <math>S</math> in a Banach space <math>X</math> is ''total'' if the [[linear span]] of <math>S</math> is [[Dense set|dense]] in <math>X.</math> The subset <math>S</math> is total in <math>X</math> if and only if the only continuous linear functional that vanishes on <math>S</math> is the <math>\mathbf{0}</math> functional: this equivalence follows from the Hahn–Banach theorem.

If <math>X</math> is the direct sum of two closed linear subspaces <math>M</math> and <math>N,</math> then the dual <math>X'</math> of <math>X</math> is isomorphic to the direct sum of the duals of <math>M</math> and <math>N.</math><ref name="Caro19">see p.&nbsp;19 in {{harvtxt|Carothers|2005}}.</ref> 
If <math>M</math> is a closed linear subspace in <math>X,</math> one can associate the {{em|orthogonal of}} <math>M</math> in the dual,
<math display=block>M^{\bot} = \{ x' \in X \mid x'(m) = 0 \text{ for all } m \in M \}.</math>

The orthogonal <math>M^{\bot}</math> is a closed linear subspace of the dual. The dual of <math>M</math> is isometrically isomorphic to <math>X' / M^{\bot}.</math> 
The dual of <math>X / M</math> is isometrically isomorphic to <math>M^{\bot}.</math><ref>Theorems&nbsp;1.10.16, 1.10.17 pp.94–95 in {{harvtxt|Megginson|1998}}</ref>

The dual of a separable Banach space need not be separable, but:

{{math theorem|name=Theorem<ref>Theorem&nbsp;1.12.11, p.&nbsp;112 in {{harvtxt|Megginson|1998}}</ref>|math_statement= Let <math>X</math> be a normed space. If <math>X'</math> is [[Separable space|separable]], then <math>X</math> is separable.}}

When <math>X'</math> is separable, the above criterion for totality can be used for proving the existence of a countable total subset in <math>X.</math>

====Weak topologies====

The ''[[weak topology]]'' on a Banach space <math>X</math> is the [[Comparison of topologies|coarsest topology]] on <math>X</math> for which all elements <math>x'</math> in the continuous dual space <math>X'</math> are continuous. 
The norm topology is therefore [[Comparison of topologies|finer]] than the weak topology. 
It follows from the Hahn–Banach separation theorem that the weak topology is [[Hausdorff space|Hausdorff]], and that a norm-closed [[Convex set|convex subset]] of a Banach space is also weakly closed.<ref>Theorem&nbsp;2.5.16, p.&nbsp;216 in {{harvtxt|Megginson|1998}}.</ref> 
A norm-continuous linear map between two Banach spaces <math>X</math> and <math>Y</math> is also ''weakly continuous'', that is, continuous from the weak topology of <math>X</math> to that of <math>Y.</math><ref>see II.A.8, p.&nbsp;29 in {{harvtxt|Wojtaszczyk|1991}}</ref>

If <math>X</math> is infinite-dimensional, there exist linear maps which are not continuous. The space <math>X^*</math> of all linear maps from <math>X</math> to the underlying field <math>\mathbb{K}</math> (this space <math>X^*</math> is called the [[Dual space#Algebraic dual space|algebraic dual space]], to distinguish it from <math>X'</math> also induces a topology on <math>X</math> which is [[finer topology|finer]] than the weak topology, and much less used in functional analysis.

On a dual space <math>X',</math> there is a topology weaker than the weak topology of <math>X',</math> called the ''[[weak topology|weak* topology]]''. 
It is the coarsest topology on <math>X'</math> for which all evaluation maps <math>x' \in X' \mapsto x'(x),</math> where <math>x</math> ranges over <math>X,</math> are continuous. 
Its importance comes from the [[Banach–Alaoglu theorem]].

{{math theorem|name=[[Banach–Alaoglu theorem]]|math_statement=Let <math>X</math> be a [[normed vector space]]. Then the [[Closed set|closed]] [[Ball (mathematics)|unit ball]] <math>B = \{x \in X \mid \|x\| \leq 1\}</math> of the dual space is [[Compact space|compact]] in the weak* topology.}}

The Banach–Alaoglu theorem can be proved using [[Tychonoff's theorem]] about infinite products of compact Hausdorff spaces. 
When <math>X</math> is separable, the unit ball <math>B'</math> of the dual is a [[Metrizable space|metrizable]] compact in the weak* topology.<ref name="DualBall">see Theorem&nbsp;2.6.23, p.&nbsp;231 in {{harvtxt|Megginson|1998}}.</ref>

====Examples of dual spaces====

The dual of <math>c_0</math> is isometrically isomorphic to <math>\ell^1</math>: for every bounded linear functional <math>f</math> on <math>c_0,</math> there is a unique element <math>y = \{y_n\} \in \ell^1</math> such that
<math display=block>f(x) = \sum_{n \in \N} x_n y_n, \qquad x = \{x_n\} \in c_0, \ \ \text{and} \ \ \|f\|_{(c_0)'} = \|y\|_{\ell_1}. </math>

The dual of <math>\ell^1</math> is isometrically isomorphic to <math>\ell^{\infty}</math>. 
The dual of [[Lp space#Properties of Lp spaces|Lebesgue space]] <math>L^p([0, 1])</math> is isometrically isomorphic to <math>L^q([0, 1])</math> when <math>1 \leq p < \infty</math> and <math>\frac{1}{p} + \frac{1}{q} = 1.</math>

For every vector <math>y</math> in a Hilbert space <math>H,</math> the mapping
<math display=block>x \in H \to f_y(x) = \langle x, y \rangle</math>

defines a continuous linear functional <math>f_y</math> on <math>H.</math>The [[Riesz representation theorem]] states that every continuous linear functional on <math>H</math> is of the form <math>f_y</math> for a uniquely defined vector <math>y</math> in <math>H.</math>
The mapping <math>y \in H \to f_y</math> is an [[Antilinear map|antilinear]] isometric bijection from <math>H</math> onto its dual <math>H'.</math> 
When the scalars are real, this map is an isometric isomorphism.

When <math>K</math> is a compact Hausdorff topological space, the dual <math>M(K)</math> of <math>C(K)</math> is the space of [[Radon measure]]s in the sense of Bourbaki.<ref>see N. Bourbaki, (2004), "Integration I", Springer Verlag, {{ISBN|3-540-41129-1}}.</ref> 
The subset <math>P(K)</math> of <math>M(K)</math> consisting of non-negative measures of mass 1 ([[probability measure]]s) is a convex w*-closed subset of the unit ball of <math>M(K).</math> 
The [[extreme point]]s of <math>P(K)</math> are the [[Dirac measure]]s on <math>K.</math> 
The set of Dirac measures on <math>K,</math> equipped with the w*-topology, is [[Homeomorphism|homeomorphic]] to <math>K.</math>

{{math theorem|name=[[Banach–Stone theorem|Banach–Stone Theorem]]|math_statement=If <math>K</math> and <math>L</math> are compact Hausdorff spaces and if <math>C(K)</math> and <math>C(L)</math> are isometrically isomorphic, then the topological spaces <math>K</math> and <math>L</math> are [[homeomorphic]].<ref name= Eilenberg /><ref>see also {{harvtxt|Banach|1932}}, p.&nbsp;170 for metrizable <math>K</math> and <math>L.</math></ref>}}

The result has been extended by Amir<ref>{{cite journal |first=Dan |last=Amir |title=On isomorphisms of continuous function spaces |journal=[[Israel Journal of Mathematics]] |volume=3 |year=1965 |issue=4 |pages=205–210 |doi=10.1007/bf03008398 |doi-access=free |s2cid=122294213 }}</ref> and Cambern<ref>{{cite journal |first=M. |last=Cambern |title=A generalized Banach–Stone theorem |journal=Proc. Amer. Math. Soc. |volume=17 |year=1966 |issue=2 |pages=396–400 |doi=10.1090/s0002-9939-1966-0196471-9|doi-access=free}} And {{cite journal |first=M. |last=Cambern |title=On isomorphisms with small bound |journal=Proc. Amer. Math. Soc. |volume=18 |year=1967 |issue=6 |pages=1062–1066 |doi=10.1090/s0002-9939-1967-0217580-2|doi-access=free}}</ref> to the case when the multiplicative [[Banach–Mazur compactum|Banach–Mazur distance]] between <math>C(K)</math> and <math>C(L)</math> is <math>< 2.</math> 
The theorem is no longer true when the distance is <math> = 2.</math><ref>{{cite journal |first=H. B. |last=Cohen |title=A bound-two isomorphism between <math>C(X)</math> Banach spaces |journal=Proc. Amer. Math. Soc. |volume=50 |year=1975 |pages=215–217 |doi=10.1090/s0002-9939-1975-0380379-5|doi-access=free }}</ref>

In the commutative [[Banach algebra]] <math>C(K),</math> the [[Banach algebra#Ideals and characters|maximal ideals]] are precisely kernels of Dirac measures on <math>K,</math>
<math display=block>I_x = \ker \delta_x = \{f \in C(K) \mid f(x) = 0\}, \quad x \in K.</math>

More generally, by the [[Gelfand–Mazur theorem]], the maximal ideals of a unital commutative Banach algebra can be identified with its [[Banach algebra#Ideals and characters|characters]]—not merely as sets but as topological spaces: the former with the [[hull-kernel topology]] and the latter with the w*-topology. 
In this identification, the maximal ideal space can be viewed as a w*-compact subset of the unit ball in the dual <math>A'.</math>

{{math theorem|math_statement= If <math>K</math> is a compact Hausdorff space, then the maximal ideal space <math>\Xi</math> of the Banach algebra <math>C(K)</math> is [[homeomorphic]] to <math>K.</math><ref name=Eilenberg>{{cite journal |last=Eilenberg |first=Samuel |title=Banach Space Methods in Topology |journal=[[Annals of Mathematics]] |date=1942 |volume=43 |issue=3 |pages=568–579 |doi=10.2307/1968812|jstor=1968812 }}</ref>}}

Not every unital commutative Banach algebra is of the form <math>C(K)</math> for some compact Hausdorff space <math>K.</math> However, this statement holds if one places <math>C(K)</math> in the smaller category of commutative [[C*-algebra]]s. 
[[Israel Gelfand|Gelfand's]] [[Gelfand representation|representation theorem]] for commutative C*-algebras states that every commutative unital ''C''*-algebra <math>A</math> is isometrically isomorphic to a <math>C(K)</math> space.<ref>See for example {{cite book |first=W. |last=Arveson |year=1976 |title=An Invitation to C*-Algebra |publisher=Springer-Verlag |isbn=0-387-90176-0 }}</ref> 
The Hausdorff compact space <math>K</math> here is again the maximal ideal space, also called the [[Spectrum of a C*-algebra#Examples|spectrum]] of <math>A</math> in the C*-algebra context.

====Bidual====
{{See also|Bidual|Reflexive space|Semi-reflexive space}}

If <math>X</math> is a normed space, the (continuous) dual <math>X''</math> of the dual <math>X'</math> is called the '''{{visible anchor|bidual}}''' or '''{{visible anchor|second dual}}''' of <math>X.</math> 
For every normed space <math>X,</math> there is a natural map,
<math display="block>\begin{cases}
F_X\colon X \to X'' \\
F_X(x) (f) = f(x) & \text{ for all } x \in X, \text{ and for all } f \in X'
\end{cases}</math>

This defines <math>F_X(x)</math> as a continuous linear functional on <math>X',</math> that is, an element of <math>X''.</math> The map <math>F_X \colon x \to F_X(x)</math> is a linear map from <math>X</math> to <math>X''.</math> 
As a consequence of the existence of a [[Banach space#Dual space|norming functional]] <math>f</math> for every <math>x \in X,</math> this map <math>F_X</math> is isometric, thus [[injective]].

For example, the dual of <math>X = c_0</math> is identified with <math>\ell^1,</math> and the dual of <math>\ell^1</math> is identified with <math>\ell^{\infty},</math> the space of bounded scalar sequences. 
Under these identifications, <math>F_X</math> is the inclusion map from <math>c_0</math> to <math>\ell^{\infty}.</math> It is indeed isometric, but not onto.

If <math>F_X</math> is [[surjective]], then the normed space <math>X</math> is called ''reflexive'' (see [[Banach space#Reflexivity|below]]). 
Being the dual of a normed space, the bidual <math>X''</math> is complete, therefore, every reflexive normed space is a Banach space.

Using the isometric embedding <math>F_X,</math> it is customary to consider a normed space <math>X</math> as a subset of its bidual. 
When <math>X</math> is a Banach space, it is viewed as a closed linear subspace of <math>X''.</math> If <math>X</math> is not reflexive, the unit ball of <math>X</math> is a proper subset of the unit ball of <math>X''.</math> 
The [[Goldstine theorem]] states that the unit ball of a normed space is weakly*-dense in the unit ball of the bidual. 
In other words, for every <math>x''</math> in the bidual, there exists a [[Net (mathematics)|net]] <math>(x_i)_{i \in I}</math> in <math>X</math> so that
<math display="block>\sup_{i \in I} \|x_i\| \leq \|x''\|, \ \ x''(f) = \lim_i f(x_i), \quad f \in X'.</math>

The net may be replaced by a weakly*-convergent sequence when the dual <math>X'</math> is separable. 
On the other hand, elements of the bidual of <math>\ell^1</math> that are not in <math>\ell^1</math> cannot be weak*-limit of {{em|sequences}} in <math>\ell^1,</math> since <math>\ell^1</math> is [[#Weak convergences of sequences|weakly sequentially complete]].

===Banach's theorems===

Here are the main general results about Banach spaces that go back to the time of Banach's book ({{harvtxt|Banach|1932}}) and are related to the [[Baire category theorem]]. 
According to this theorem, a complete metric space (such as a Banach space, a [[Fréchet space]] or an [[F-space]]) cannot be equal to a union of countably many closed subsets with empty [[Interior (topology)|interiors]]. 
Therefore, a Banach space cannot be the union of countably many closed subspaces, unless it is already equal to one of them; a Banach space with a countable [[Hamel basis]] is finite-dimensional.

{{math theorem|name=[[Uniform boundedness principle|Banach–Steinhaus Theorem]]|math_statement=Let <math>X</math> be a Banach space and <math>Y</math> be a [[normed vector space]]. Suppose that <math>F</math> is a collection of continuous linear operators from <math>X</math> to <math>Y.</math> The uniform boundedness principle states that if for all <math>x</math> in <math>X</math> we have <math>\sup_{T \in F} \|T(x)\|_Y < \infty,</math> then <math>\sup_{T \in F} \|T\|_Y < \infty.</math>}}

The Banach–Steinhaus theorem is not limited to Banach spaces. 
It can be extended for example to the case where <math>X</math> is a [[Fréchet space]], provided the conclusion is modified as follows: under the same hypothesis, there exists a neighborhood <math>U</math> of <math>\mathbf{0}</math> in <math>X</math> such that all <math>T</math> in <math>F</math> are uniformly bounded on <math>U,</math>
<math display=block>\sup_{T \in F} \sup_{x \in U} \; \|T(x)\|_Y < \infty.</math>

{{math theorem|name=[[Open mapping theorem (functional analysis)|The Open Mapping Theorem]]|math_statement=Let <math>X</math> and <math>Y</math> be Banach spaces and <math>T : X \to Y</math> be a surjective continuous linear operator, then <math>T</math> is an open map.}}

{{math theorem|name=Corollary | math_statement = Every one-to-one bounded linear operator from a Banach space onto a Banach space is an isomorphism.}}

{{math theorem|name=The First Isomorphism Theorem for Banach spaces | math_statement= Suppose that <math>X</math> and <math>Y</math> are Banach spaces and that <math>T \in B(X, Y).</math> Suppose further that the range of <math>T</math> is closed in <math>Y.</math> Then <math>X / \ker T</math> is isomorphic to <math>T(X).</math>}}

This result is a direct consequence of the preceding ''Banach isomorphism theorem'' and of the canonical factorization of bounded linear maps.

{{math theorem|name=Corollary|math_statement=If a Banach space <math>X</math> is the internal direct sum of closed subspaces <math>M_1, \ldots, M_n,</math> then <math>X</math> is isomorphic to <math>M_1 \oplus \cdots \oplus M_n.</math>}}

This is another consequence of Banach's isomorphism theorem, applied to the continuous bijection from <math>M_1 \oplus \cdots \oplus M_n</math> onto <math>X</math> sending <math>m_1, \cdots, m_n</math> to the sum <math>m_1 + \cdots + m_n.</math>

{{math theorem|name=[[Closed graph theorem|The Closed Graph Theorem]]|math_statement= Let <math>T : X \to Y</math> be a linear mapping between Banach spaces. The graph of <math>T</math> is closed in <math>X \times Y</math> if and only if <math>T</math> is continuous.}}

===Reflexivity===
{{main|Reflexive space}}

The normed space <math>X</math> is called ''[[Reflexive space|reflexive]]'' when the natural map
<math display=block>\begin{cases} F_X : X \to X'' \\ F_X(x) (f) = f(x) & \text{ for all } x \in X, \text{ and for all } f \in X'\end{cases}</math>
is surjective. Reflexive normed spaces are Banach spaces.

{{math theorem| math_statement = If <math>X</math> is a reflexive Banach space, every closed subspace of <math>X</math> and every quotient space of <math>X</math> are reflexive.}}

This is a consequence of the Hahn–Banach theorem. 
Further, by the open mapping theorem, if there is a bounded linear operator from the Banach space <math>X</math> onto the Banach space <math>Y,</math> then <math>Y</math> is reflexive.

{{math theorem| math_statement = If <math>X</math> is a Banach space, then <math>X</math> is reflexive if and only if <math>X'</math> is reflexive.}}

{{math theorem|name=Corollary | math_statement = Let <math>X</math> be a reflexive Banach space. Then <math>X</math> is [[Separable space|separable]] if and only if <math>X'</math> is separable.}}

Indeed, if the dual <math>Y'</math> of a Banach space <math>Y</math> is separable, then <math>Y</math> is separable. 
If <math>X</math> is reflexive and separable, then the dual of <math>X'</math> is separable, so <math>X'</math> is separable.

{{math theorem| math_statement = Suppose that <math>X_1, \ldots, X_n</math> are normed spaces and that <math>X = X_1 \oplus \cdots \oplus X_n.</math> Then <math>X</math> is reflexive if and only if each <math>X_j</math> is reflexive.}}

Hilbert spaces are reflexive. The <math>L^p</math> spaces are reflexive when <math>1 < p < \infty.</math> More generally, [[uniformly convex space]]s are reflexive, by the [[Milman–Pettis theorem]]. 
The spaces <math>c_0, \ell^1, L^1([0, 1]), C([0, 1])</math> are not reflexive. 
In these examples of non-reflexive spaces <math>X,</math> the bidual <math>X''</math> is "much larger" than <math>X.</math> 
Namely, under the natural isometric embedding of <math>X</math> into <math>X''</math> given by the Hahn–Banach theorem, the quotient <math>X'' / X</math> is infinite-dimensional, and even nonseparable. 
However, Robert C. James has constructed an example<ref>{{cite journal|author = R. C. James|title=A non-reflexive Banach space isometric with its second conjugate space|journal=Proc. Natl. Acad. Sci. U.S.A.|volume=37|pages=174–177|year=1951|issue=3 | doi=10.1073/pnas.37.3.174 | pmc=1063327|pmid=16588998|bibcode=1951PNAS...37..174J |doi-access=free}}</ref> of a non-reflexive space, usually called "''the James space''" and denoted by <math>J,</math><ref>see {{harvtxt|Lindenstrauss|Tzafriri|1977}}, p.&nbsp;25.</ref> such that the quotient <math>J'' / J</math> is one-dimensional. 
Furthermore, this space <math>J</math> is isometrically isomorphic to its bidual.

{{math theorem| math_statement = A Banach space <math>X</math> is reflexive if and only if its unit ball is [[Compact space|compact]] in the [[weak topology]].}}

When <math>X</math> is reflexive, it follows that all closed and bounded [[Convex set|convex subsets]] of <math>X</math> are weakly compact. 
In a Hilbert space <math>H,</math> the weak compactness of the unit ball is very often used in the following way: every bounded sequence in <math>H</math> has weakly convergent subsequences.

Weak compactness of the unit ball provides a tool for finding solutions in reflexive spaces to certain [[Infinite-dimensional optimization|optimization problems]]. 
For example, every [[Convex function|convex]] continuous function on the unit ball <math>B</math> of a reflexive space attains its minimum at some point in <math>B.</math>

As a special case of the preceding result, when <math>X</math> is a reflexive space over <math>\R,</math> every continuous linear functional <math>f</math> in <math>X'</math> attains its maximum <math>\|f\|</math> on the unit ball of <math>X.</math> 
The following [[James' theorem|theorem of Robert C. James]] provides a converse statement.

{{math theorem| name = James' Theorem | math_statement = For a Banach space the following two properties are equivalent:
* <math>X</math> is reflexive.
* for all <math>f</math> in <math>X'</math> there exists <math>x \in X</math> with <math>\|x\| \leq 1,</math> so that <math>f(x) = \|f\|.</math>}}

The theorem can be extended to give a characterization of weakly compact convex sets.

On every non-reflexive Banach space <math>X,</math> there exist continuous linear functionals that are not ''norm-attaining''. 
However, the [[Errett Bishop|Bishop]]–[[Robert Phelps|Phelps]] theorem<ref>{{cite journal|last1=bishop|first1=See E.|last2=Phelps|first2=R.|year=1961|title=A proof that every Banach space is subreflexive|journal=Bull. Amer. Math. Soc.|volume=67|pages=97–98|doi=10.1090/s0002-9904-1961-10514-4|doi-access=free }}</ref> states that norm-attaining functionals are norm dense in the dual <math>X'</math> of <math>X.</math>

===Weak convergences of sequences===

A sequence <math>\{x_n\}</math> in a Banach space <math>X</math> is ''weakly convergent'' to a vector <math>x \in X</math> if <math>\{f(x_n)\}</math> converges to <math>f(x)</math> for every continuous linear functional <math>f</math> in the dual <math>X'.</math> The sequence <math>\{x_n\}</math> is a ''weakly Cauchy sequence'' if <math>\{f(x_n)\}</math> converges to a scalar limit <math>L(f)</math> for every <math>f</math> in <math>X'.</math> 
A sequence <math>\{f_n\}</math> in the dual <math>X'</math> is ''weakly* convergent'' to a functional <math>f \in X'</math> if <math>f_n(x)</math> converges to <math>f(x)</math> for every <math>x</math> in <math>X.</math> 
Weakly Cauchy sequences, weakly convergent and weakly* convergent sequences are norm bounded, as a consequence of the [[Uniform boundedness principle|Banach–Steinhaus]] theorem.

When the sequence <math>\{x_n\}</math> in <math>X</math> is a weakly Cauchy sequence, the limit <math>L</math> above defines a bounded linear functional on the dual <math>X',</math> that is, an element <math>L</math> of the bidual of <math>X,</math> and <math>L</math> is the limit of <math>\{x_n\}</math> in the weak*-topology of the bidual. 
The Banach space <math>X</math> is ''weakly sequentially complete'' if every weakly Cauchy sequence is weakly convergent in <math>X.</math> 
It follows from the preceding discussion that reflexive spaces are weakly sequentially complete.

{{math theorem| name = Theorem <ref>see III.C.14, p.&nbsp;140 in {{harvtxt|Wojtaszczyk|1991}}.</ref> | math_statement = For every measure <math>\mu,</math> the space <math>L^1(\mu)</math> is weakly sequentially complete.}}

An orthonormal sequence in a Hilbert space is a simple example of a weakly convergent sequence, with limit equal to the <math>\mathbf{0}</math> vector. 
The [[Schauder basis#Examples|unit vector basis]] of <math>\ell^p</math> for <math>1 < p < \infty,</math> or of <math>c_0,</math> is another example of a ''weakly null sequence'', that is, a sequence that converges weakly to <math>\mathbf{0}.</math> 
For every weakly null sequence in a Banach space, there exists a sequence of convex combinations of vectors from the given sequence that is norm-converging to <math>\mathbf{0}.</math><ref>see Corollary&nbsp;2, p.&nbsp;11 in {{harvtxt|Diestel|1984}}.</ref>

The unit vector basis of <math>\ell^1</math> is not weakly Cauchy. 
Weakly Cauchy sequences in <math>\ell^1</math> are weakly convergent, since <math>L^1</math>-spaces are weakly sequentially complete. 
Actually, weakly convergent sequences in <math>\ell^1</math> are norm convergent.<ref>see p.&nbsp;85 in {{harvtxt|Diestel|1984}}.</ref> This means that <math>\ell^1</math> satisfies [[Schur's property]].

====Results involving the {{math|𝓁<sup>1</sup>}} basis====

Weakly Cauchy sequences and the <math>\ell^1</math> basis are the opposite cases of the dichotomy established in the following deep result of&nbsp;H.&nbsp;P.&nbsp;Rosenthal.<ref>{{cite journal|last1=Rosenthal|first1=Haskell P|year=1974|title=A characterization of Banach spaces containing ℓ<sup>1</sup>|journal=Proc. Natl. Acad. Sci. U.S.A.|volume=71|issue=6| pages=2411–2413 | doi=10.1073/pnas.71.6.2411|pmid=16592162|pmc=388466|arxiv=math.FA/9210205|bibcode=1974PNAS...71.2411R|doi-access=free}} Rosenthal's proof is for real scalars. The complex version of the result is due to L. Dor, in {{cite journal| last1=Dor|first1=Leonard E|year=1975|title=On sequences spanning a complex ℓ<sup>1</sup> space|journal=Proc. Amer. Math. Soc. | volume=47|pages=515–516|doi=10.1090/s0002-9939-1975-0358308-x|doi-access=free}}</ref>

{{math theorem| name = Theorem<ref>see p.&nbsp;201 in {{harvtxt|Diestel|1984}}.</ref> | math_statement = Let <math>\{x_n\}_{n \in \N}</math> be a bounded sequence in a Banach space. Either <math>\{x_n\}_{n \in \N}</math> has a weakly Cauchy subsequence, or it admits a subsequence [[Schauder basis#Definitions|equivalent]] to the standard unit vector basis of <math>\ell^1.</math>}}

A complement to this result is due to Odell and Rosenthal&nbsp;(1975).

{{math theorem| name = Theorem<ref>{{citation|last1=Odell|first1=Edward W.|last2=Rosenthal|first2=Haskell P.|title=A double-dual characterization of separable Banach spaces containing ℓ<sup>1</sup>|journal=[[Israel Journal of Mathematics]]|volume=20|year=1975|issue=3–4 |pages=375–384|doi=10.1007/bf02760341|doi-access=free|s2cid=122391702|url=http://dml.cz/bitstream/handle/10338.dmlcz/133414/CommentatMathUnivCarolRetro_50-2009-1_5.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://dml.cz/bitstream/handle/10338.dmlcz/133414/CommentatMathUnivCarolRetro_50-2009-1_5.pdf |archive-date=2022-10-09 |url-status=live}}.</ref> | math_statement = Let <math>X</math> be a separable Banach space. The following are equivalent:
*The space <math>X</math> does not contain a closed subspace isomorphic to <math>\ell^1.</math>
*Every element of the bidual <math>X''</math> is the weak*-limit of a sequence <math>\{x_n\}</math> in <math>X.</math>}}

By the Goldstine theorem, every element of the unit ball <math>B''</math> of <math>X''</math> is weak*-limit of a net in the unit ball of <math>X.</math> When <math>X</math> does not contain <math>\ell^1,</math> every element of <math>B''</math> is weak*-limit of a {{em|sequence}} in the unit ball of <math>X.</math><ref>Odell and Rosenthal, Sublemma p.&nbsp;378 and Remark p.&nbsp;379.</ref>

When the Banach space <math>X</math> is separable, the unit ball of the dual <math>X',</math> equipped with the weak*-topology, is a metrizable compact space <math>K,</math><ref name="DualBall" /> and every element <math>x''</math> in the bidual <math>X''</math> defines a bounded function on <math>K</math>:
<math display=block>x' \in K \mapsto x''(x'), \quad |x''(x')| \leq \|x''\|.</math>

This function is continuous for the compact topology of <math>K</math> if and only if <math>x''</math> is actually in <math>X,</math> considered as subset of <math>X''.</math> 
Assume in addition for the rest of the paragraph that <math>X</math> does not contain <math>\ell^1.</math> 
By the preceding result of Odell and Rosenthal, the function <math>x''</math> is the [[Pointwise convergence|pointwise limit]] on <math>K</math> of a sequence <math>\{x_n\} \subseteq X</math> of continuous functions on <math>K,</math> it is therefore a [[Baire function|first Baire class function]] on <math>K.</math> 
The unit ball of the bidual is a pointwise compact subset of the first Baire class on <math>K.</math><ref>for more on pointwise compact subsets of the Baire class, see {{citation|last1=Bourgain|first1=Jean|author1-link=Jean Bourgain|last2=Fremlin|first2=D. H.|last3=Talagrand |first3=Michel|title=Pointwise Compact Sets of Baire-Measurable Functions|journal=Am. J. Math.|volume=100|year=1978|issue=4|pages=845–886|jstor=2373913|doi=10.2307/2373913}}.</ref>

====Sequences, weak and weak* compactness====

When <math>X</math> is separable, the unit ball of the dual is weak*-compact by the [[Banach–Alaoglu theorem]] and metrizable for the weak* topology,<ref name="DualBall" /> hence every bounded sequence in the dual has weakly* convergent subsequences. 
This applies to separable reflexive spaces, but more is true in this case, as stated below.

The weak topology of a Banach space <math>X</math> is metrizable if and only if <math>X</math> is finite-dimensional.<ref>see Proposition&nbsp;2.5.14, p.&nbsp;215 in {{harvtxt|Megginson|1998}}.</ref> If the dual <math>X'</math> is separable, the weak topology of the unit ball of <math>X</math> is metrizable. 
This applies in particular to separable reflexive Banach spaces. 
Although the weak topology of the unit ball is not metrizable in general, one can characterize weak compactness using sequences.

{{math theorem| name = [[Eberlein–Šmulian theorem]]<ref>see for example p.&nbsp;49, II.C.3 in {{harvtxt|Wojtaszczyk|1991}}.</ref> | math_statement = A set <math>A</math> in a Banach space is relatively weakly compact if and only if every sequence <math>\{a_n\}</math> in <math>A</math> has a weakly convergent subsequence.}}

A Banach space <math>X</math> is reflexive if and only if each bounded sequence in <math>X</math> has a weakly convergent subsequence.<ref>see Corollary&nbsp;2.8.9, p.&nbsp;251 in {{harvtxt|Megginson|1998}}.</ref>

A weakly compact subset <math>A</math> in <math>\ell^1</math> is norm-compact. Indeed, every sequence in <math>A</math> has weakly convergent subsequences by Eberlein–Šmulian, that are norm convergent by the Schur property of <math>\ell^1.</math>

=== Type and cotype ===
{{main|Type and cotype of a Banach space}}
A way to classify Banach spaces is through the probabilistic notion of [[Type and cotype of a Banach space|type and cotype]], these two measure how far a Banach space is from a Hilbert space.