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
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$\ell^p$ 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.
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$L^p$ 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.
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$C([a,b])$ (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.
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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.
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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.