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: $$\ell^p = \left\{ (x_n)_{n\in \mathbb{N}} \mid \sum_{n=1}^{\infty} |x_n|^p < \infty \right\}, \quad 1 \leq p < \infty$$
   • The norm is given by: $$\|x\|_p = \left( \sum_{n=1}^{\infty} |x_n|^p \right)^{1/p}$$
   • When $p = \infty$, we define: $$\ell^\infty = \left\{ (x_n)_{n\in \mathbb{N}} \mid \sup_n |x_n| < \infty \right\}$$ with the norm $\|x\|_{\infty} = \sup_n |x_n|$.
   • 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: $$\|f\|_p = \left( \int_X |f(x)|^p d\mu(x) \right)^{1/p} < \infty, \quad 1 \leq p < \infty.$$
   • When $p = \infty$, the norm is: $$\|f\|_{\infty} = \text{ess sup} |f(x)|.$$
   • 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: $$\|f\|_{\infty} = \sup_{x \in [a,b]} |f(x)|$$
   • 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: $$\|u\|_{W^{k,p}} = \sum_{|\alpha| \leq k} \|\partial^\alpha u\|_p.$$
   • 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: $$\|T\| = \sup_{\|x\|\leq 1} \|T(x)\|$$
   • This forms a Banach space.