Banach Spaces

What are Banach Spaces?

A Banach space is a complete normed vector space, meaning that every Cauchy sequence in the space converges to a limit within the space. See Spazi vettoriali for the formal definition.

Examples of Banach Spaces

In this section, we list some examples of the most common Banach Spaces

4. ${\ell }^p$ Spaces (Sequence Spaces)

   • Defined as: $${\ell }^p=\left\{(x_n{)}_{n\in \mathbb{N}}\mid \sum _{n=1}^{\infty }|x_n{|}^p<\infty \right\},1\le p<\infty$$
   • The norm is given by: $$\Vert x{\Vert }_p={\left(\sum _{n=1}^{\infty }|x_n{|}^p\right)}^{1/p}$$
   • When $p=\infty$, we define: $${\ell }^{\infty }=\left\{(x_n{)}_{n\in \mathbb{N}}\mid \underset{n}{\sup }|x_n|<\infty \right\}$$ with the norm $\Vert x{\Vert }_{\infty }={\sup }_n|x_n|$.
   • These spaces are Banach under their respective norms.

5. $L^p$ Spaces (Function Spaces)

   • Given a measure space $(X,\Sigma ,\mu )$, the space $L^p(X)$ consists of measurable functions $f:X\to \mathbb{R}$ (or $\mathbb{C}$) such that: $$\Vert f{\Vert }_p={\left({\int }_X|f(x){|}^{p}d\mu (x)\right)}^{1/p}<\infty ,1\le p<\infty .$$
   • When $p=\infty$, the norm is: $$\Vert f{\Vert }_{\infty }=\text{ess sup}|f(x)|.$$
   • These are Banach spaces when equipped with their respective norms.

6. $C([a,b])$ (Space of Continuous Functions)

   • The space of continuous functions on a closed interval $[a,b]$ with the supremum norm: $$\Vert f{\Vert }_{\infty }=\underset{x\in [a,b]}{\sup }|f(x)|$$
   • This is a Banach space because uniform limits of continuous functions are continuous.

7. Sobolev Spaces $W^{k,p}(\Omega )$

   • These spaces generalize $L^p$ spaces by incorporating weak derivatives up to order $k$, with norms: $$\Vert u{\Vert }_{W^{k,p}}=\sum _{|\alpha |\le k}\Vert {\partial }^{\alpha }u{\Vert }_p.$$
   • They play a crucial role in PDEs and functional analysis.

8. The Space of Bounded Linear Operators $B(X,Y)$

   • Given two Banach spaces $X$ and $Y$, the space of bounded linear operators from $X$ to $Y$, denoted $B(X,Y)$, with the operator norm: $$\Vert T\Vert =\underset{\Vert x\Vert \le 1}{\sup }\Vert T(x)\Vert$$
   • This forms a Banach space.