Editing Banach space (section)

===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.