Editing Banach space (section)

==Some classification results==

===Characterizations of Hilbert space among Banach spaces===

A necessary and sufficient condition for the norm of a Banach space <math>X</math> to be associated to an inner product is the [[parallelogram identity]]:

{{math theorem| name = Parallelogram identity | math_statement = for all <math>x, y \in X : \qquad \|x+y\|^2 + \|x-y\|^2 = 2(\|x\|^2 + \|y\|^2).</math>}}

It follows, for example, that the [[Lp space|Lebesgue space]] <math>L^p([0, 1])</math> is a Hilbert space only when <math>p = 2.</math> 
If this identity is satisfied, the associated inner product is given by the [[polarization identity]]. In the case of real scalars, this gives:
<math display=block>\langle x, y\rangle = \tfrac{1}{4}(\|x+y\|^2 - \|x-y\|^2).</math>

For complex scalars, defining the [[Inner product space|inner product]] so as to be <math>\Complex</math>-linear in <math>x,</math> [[Antilinear map|antilinear]] in <math>y,</math> the polarization identity gives:
<math display=block>\langle x,y\rangle = \tfrac{1}{4}\left(\|x+y\|^2 - \|x-y\|^2 + i(\|x+iy\|^2 - \|x-iy\|^2)\right).</math>

To see that the parallelogram law is sufficient, one observes in the real case that <math>\langle x, y \rangle</math> is symmetric, and in the complex case, that it satisfies the [[Hermitian symmetry]] property and <math>\langle i x, y \rangle = i \langle x, y \rangle.</math> The parallelogram law implies that <math>\langle x, y \rangle</math> is additive in <math>x.</math> 
It follows that it is linear over the rationals, thus linear by continuity.

Several characterizations of spaces isomorphic (rather than isometric) to Hilbert spaces are available. 
The parallelogram law can be extended to more than two vectors, and weakened by the introduction of a two-sided inequality with a constant <math>c \geq 1</math>: Kwapień proved that if
<math display=block>c^{-2} \sum_{k=1}^n \|x_k\|^2 \leq \operatorname{Ave}_{\pm} \left\|\sum_{k=1}^n \pm x_k\right\|^2 \leq c^2 \sum_{k=1}^n \|x_k\|^2</math>
for every integer <math>n</math> and all families of vectors <math>\{x_1, \ldots, x_n\} \subseteq X,</math> then the Banach space <math>X</math> is isomorphic to a Hilbert space.<ref>see Kwapień, S. (1970), "A linear topological characterization of inner-product spaces", Studia Math. '''38''':277–278.</ref> 
Here, <math>\operatorname{Ave}_{\pm}</math> denotes the average over the <math>2^n</math> possible choices of signs <math>\pm 1.</math>
In the same article, Kwapień proved that the validity of a Banach-valued [[Parseval's theorem]] for the Fourier transform characterizes Banach spaces isomorphic to Hilbert spaces.

Lindenstrauss and Tzafriri proved that a Banach space in which every closed linear subspace is complemented (that is, is the range of a bounded linear projection) is isomorphic to a Hilbert space.<ref>{{cite journal
|last1=Lindenstrauss|first1=Joram
|last2=Tzafriri|first2=Lior
|year=1971
|title=On the complemented subspaces problem
|journal=[[Israel Journal of Mathematics]]
|volume=9
|issue=2
|pages=263–269
|doi=10.1007/BF02771592 | doi-access=free}}</ref> The proof rests upon [[Dvoretzky's theorem]] about Euclidean sections of high-dimensional centrally symmetric convex bodies. In other words, Dvoretzky's theorem states that for every integer <math>n,</math> any finite-dimensional normed space, with dimension sufficiently large compared to <math>n,</math> contains subspaces nearly isometric to the <math>n</math>-dimensional Euclidean space.

The next result gives the solution of the so-called {{em|homogeneous space problem}}. An infinite-dimensional Banach space <math>X</math> is said to be ''homogeneous'' if it is isomorphic to all its infinite-dimensional closed subspaces. A Banach space isomorphic to <math>\ell^2</math> is homogeneous, and Banach asked for the converse.<ref>see p.&nbsp;245 in {{harvtxt|Banach|1932}}. The homogeneity property is called "propriété&nbsp;(15)" there. Banach writes: "on ne connaît aucun exemple d'espace à une infinité de dimensions qui, sans être isomorphe avec <math>(L^2).</math> possède la propriété&nbsp;(15)".</ref>

{{math theorem| name = Theorem<ref name="Gowers">Gowers, W. T. (1996), "A new dichotomy for Banach spaces", Geom. Funct. Anal. '''6''':1083–1093.</ref> | math_statement = A Banach space isomorphic to all its infinite-dimensional closed subspaces is isomorphic to a separable Hilbert space.}}

An infinite-dimensional Banach space is ''hereditarily indecomposable'' when no subspace of it can be isomorphic to the direct sum of two infinite-dimensional Banach spaces. 
The [[Timothy Gowers|Gowers]] dichotomy theorem<ref name="Gowers" /> asserts that every infinite-dimensional Banach space <math>X</math> contains, either a subspace <math>Y</math> with [[Schauder basis#Unconditionality|unconditional basis]], or a hereditarily indecomposable subspace <math>Z,</math> and in particular, <math>Z</math> is not isomorphic to its closed hyperplanes.<ref>see {{cite journal|last1=Gowers|first1=W. T.|year=1994|title=A solution to Banach's hyperplane problem|journal=Bull. London Math. Soc.|volume=26|issue=6|pages=523–530|doi=10.1112/blms/26.6.523}}</ref> 
If <math>X</math> is homogeneous, it must therefore have an unconditional basis. It follows then from the partial solution obtained by Komorowski and [[Nicole Tomczak-Jaegermann|Tomczak–Jaegermann]], for spaces with an unconditional basis,<ref>see {{cite journal|last1=Komorowski|first1=Ryszard A.|last2=Tomczak-Jaegermann|first2=Nicole|year=1995|title=Banach spaces without local unconditional structure|journal=[[Israel Journal of Mathematics]]|volume=89|issue=1–3|pages=205–226|arxiv=math/9306211|doi=10.1007/bf02808201|doi-access=free|s2cid=5220304}} and also {{cite journal|last1=Komorowski|first1=Ryszard A.|last2=Tomczak-Jaegermann|first2=Nicole|year=1998|title=Erratum to: Banach spaces without local unconditional structure|journal=[[Israel Journal of Mathematics]]|volume=105|pages=85–92|arxiv=math/9607205|doi=10.1007/bf02780323|doi-access=free|s2cid=18565676}}</ref> that <math>X</math> is isomorphic to <math>\ell^2.</math>

===Metric classification===

If <math>T : X \to Y</math> is an [[isometry]] from the Banach space <math>X</math> onto the Banach space <math>Y</math> (where both <math>X</math> and <math>Y</math> are vector spaces over <math>\R</math>), then the [[Mazur–Ulam theorem]] states that <math>T</math> must be an affine transformation. 
In particular, if <math>T(0_X) = 0_Y,</math> this is <math>T</math> maps the zero of <math>X</math> to the zero of <math>Y,</math> then <math>T</math> must be linear. This result implies that the metric in Banach spaces, and more generally in normed spaces, completely captures their linear structure.

===Topological classification===

Finite dimensional Banach spaces are homeomorphic as topological spaces, if and only if they have the same dimension as real vector spaces.

[[Anderson–Kadec theorem]] (1965–66) proves<ref>{{cite book|author=C. Bessaga, A. Pełczyński|title=Selected Topics in Infinite-Dimensional Topology|url=https://books.google.com/books?id=7n9sAAAAMAAJ|year=1975|publisher=Panstwowe wyd. naukowe|pages=177–230}}</ref> that any two infinite-dimensional [[separable space|separable]] Banach spaces are homeomorphic as topological spaces. Kadec's theorem was extended by Torunczyk, who proved<ref>{{cite book |author=H. Torunczyk |title=Characterizing Hilbert Space Topology |publisher=Fundamenta Mathematicae |year=1981 |pages=247–262}}</ref> that any two Banach spaces are homeomorphic if and only if  they have the same [[Set-theoretic topology#Cardinal functions|density character]], the minimum cardinality of a dense subset.

===Spaces of continuous functions===

When two compact Hausdorff spaces <math>K_1</math> and <math>K_2</math> are [[Homeomorphism|homeomorphic]], the Banach spaces <math>C(K_1)</math> and <math>C(K_2)</math> are isometric. Conversely, when <math>K_1</math> is not homeomorphic to <math>K_2,</math> the (multiplicative) Banach–Mazur distance between <math>C(K_1)</math> and <math>C(K_2)</math> must be greater than or equal to <math>2,</math> see above the [[#Examples of dual spaces|results by Amir and Cambern]]. 
Although uncountable compact metric spaces can have different homeomorphy types, one has the following result due to Milutin:<ref>Milyutin, Alekseĭ A. (1966), "Isomorphism of the spaces of continuous functions over compact sets of the cardinality of the continuum". (Russian) Teor. Funkciĭ Funkcional. Anal. i Priložen. Vyp. '''2''':150–156.</ref>

{{math theorem| name = Theorem<ref>Milutin. See also Rosenthal, Haskell P., "The Banach spaces C(K)" in Handbook of the geometry of Banach spaces, Vol. 2, 1547–1602, North-Holland, Amsterdam, 2003.</ref> | math_statement =Let <math>K</math> be an uncountable compact metric space. Then <math>C(K)</math> is isomorphic to  <math>C([0, 1]).</math>}}

The situation is different for [[Countable set|countably infinite]] compact Hausdorff spaces. 
Every countably infinite compact <math>K</math> is homeomorphic to some closed interval of [[ordinal number]]s
<math display=block>\langle 1, \alpha \rangle = \{ \gamma \mid 1 \leq \gamma \leq \alpha\}</math>
equipped with the [[order topology]], where <math>\alpha</math> is a countably infinite ordinal.<ref>One can take {{math|1=''α'' = ''ω''{{i sup|''βn''}}}}, where <math>\beta + 1</math> is the [[Derived set (mathematics)#Cantor–Bendixson rank|Cantor–Bendixson rank]] of <math>K,</math> and <math>n > 0</math> is the finite number of points in the <math>\beta</math>-th [[Derived set (mathematics)|derived set]] <math>K(\beta)</math> of <math>K.</math> See [[Stefan Mazurkiewicz|Mazurkiewicz, Stefan]]; [[Wacław Sierpiński|Sierpiński, Wacław]] (1920), "Contribution à la topologie des ensembles dénombrables", Fundamenta Mathematicae 1: 17–27.</ref> 
The Banach space <math>C(K)</math> is then isometric to {{math|''C''(⟨1, ''α''⟩)}}. When <math>\alpha, \beta</math> are two countably infinite ordinals, and assuming <math>\alpha \leq \beta,</math> the spaces {{math|''C''(⟨1, ''α''⟩)}} and {{math|''C''(⟨1, ''β''⟩)}} are isomorphic if and only if {{math|''β'' < ''α<sup>ω</sup>''}}.<ref>Bessaga, Czesław; Pełczyński, Aleksander (1960), "Spaces of continuous functions. IV. On isomorphical classification of spaces of continuous functions", Studia Math. '''19''':53–62.</ref>
For example, the Banach spaces
<math display=block>C(\langle 1, \omega\rangle), \ C(\langle 1, \omega^{\omega} \rangle), \ C(\langle 1, \omega^{\omega^2}\rangle), \ C(\langle 1, \omega^{\omega^3} \rangle), \cdots, C(\langle 1, \omega^{\omega^\omega} \rangle), \cdots</math>
are mutually non-isomorphic.