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Functional Analysis 1 -- Banach Space

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Functional Analysis 1 -- Banach Space

This is course note from Siran Li-Shanghai Jiao Tong University.

Definition: Banach Spaces

Let \(X\) be a vector space over \(\mathbb{R}\) or \(\mathbb{C}\). A norm on \(X\) is a function \(||\cdot||:X\rightarrow [0,\infty[\) such that:

  1. \(||x|| = 0 \Rightarrow x = 0\).
  2. \(||\lambda x|| = |\lambda| ||x||, \forall x \in X, \lambda \in \mathbb{K}\).
  3. \(||x+y|| \le ||x|| + ||y||, \forall x,y \in X\).

A norm on \(X\) gives rise to a metric(distance).

Definition: Metric

Let \(X\) be a topological space. A function \(d: X \times X \rightarrow [0,\infty[\) is a metric if

  1. \(d(x,y) = 0 \Rightarrow x = y\).
  2. \(d(x,y) = d(y,x)\)
  3. \(d(x,y) \le d(x,z)+d(z,y)\)

Let \((X, ||\cdot||)\) be a normed vector space(NVS), then \(d(x,y) := ||x-y||\) defines a metric on \(X\).

Remark: \((X, ||\cdot||)\) is a metric space \(\neq\) the metric is from the norm.

Let \((X, ||\cdot||)\) be a NVS with \(d(x,y) = ||x-y||\). It satisfies the following geometric properties:

  1. Translation Invariance: \(d(x+z, y+z) = d(x, y)\).
  2. One-Homogeneity: \(\quad\) \(d(\lambda x, \lambda y) = |\lambda| d(x,y)\).
  3. Open balls are convex: \(B^{\circ}(x_0, \rho) := \{x \in X: d(x, x_0) < \rho\}\)

The topology on NVS that arises from \(d(x,y) = ||x-y||\) is referenced to as the norm topology.

Definition: Cauthy

Let \((X, ||\cdot||)\) be a NVS. A sequence \(\{x_n\} \subset X\) is Cauthy iff. \(||x_n - x_l|| \rightarrow 0\) as \(n,l \rightarrow \infty\).

Example: \(\{\frac{1}{n}\}\) is Cauthy in \(\mathbb{R} \backslash \{0\}\) but not convergent in \(\mathbb{R} \backslash \{0\}\).

Definition: Complete

Let \((X, ||\cdot||)\) be a NVS. The norm is complete iff every Cauthy sequence is convergent in X.

Definition: Banach Space

A complete NVS is called a Banach space.

Examples:

  1. \((\mathbb{R}, ||\cdot||_p)\) for \(1 \le p \le \infty\).
  2. \(l^p(\mathbb{K}) = \{ \{a_n\}_{n \in \mathbb{N}}: (\sum_{n=1}^{\infty} |a_n|^p)^{1/p} < \infty\}\).
  3. \(l^{\infty}(\mathbb{K}) = \{ \{a_n\}_{n \in \mathbb{N}}: \underset{n\in \mathbb{N}}{max} |a_n| < \infty\}\).

Remark: Differentiability of \(||\cdot|| \leftrightarrow\) Uniform convexity of unit ball.

Definition: Unit Ball

Let \(\mathbb{S}^d_p = \{x \in \mathbb{R}^d: ||x||_p = 1\}\). It represents unit ball in \(d\) dimension space with \(l_p\) norm.

Then we have the following properties for \(\mathbb{S}^d_p\):

  1. \(\mathbb{S}^d_p \rightarrow \mathbb{S}^d_1\) as \(p \rightarrow 1\).
  2. \(\mathbb{S}^d_p \rightarrow \mathbb{S}^d_{\infty}\) as \(p \rightarrow \infty\).

Remark: in infinite dimensional space, compact \(\neq\) closed + bounded. Example: Let \(K = [0,1]\). The sequence \(\{f_n\}\) with \(f_n(x) = x^n\) satisfies:

  1. \(||f_n||_{\infty} = 1, \forall n\).
  2. \(f_n\) does not have a convergent subsequence.

Theorem1.1: > Let \((X, ||\cdot||)\) be a Banach space. Let \(Y \le X\) be a vetor subspace. Then \((Y, ||\cdot||)\) is a Banach space \(\Leftrightarrow\) \(Y\) is closed in \(X\).

proof: \(\Rightarrow\) Assume that \((Y, ||\cdot||)\) is a Banach space . Take any \(\{y_n\} \subset Y\), s.t. \(y_n \rightarrow x \in X\). Hence \(\{y_n\}\) is Cauchy, so \(\{y_n\}\) converges in \(Y\). By uniqueness of limit in metric space, \(x \in Y\). Therefore, \(Y\) is closed.

\(\Leftarrow\) Assume that \(Y\) is closed. Let \(\{y_n\}\) be Cauthy in \(Y\). Hence \(\{y_n\}\) is Cauchy in \(X\). But \(X\) is a Banach space, then \(y_n \rightarrow x \in X\). Thus, \(x \in Y\) as \(Y\) is closed. Thus, \(Y\) is a Banach space. \(\Box\)

Example: Let \(Z = span \{e_n \mid n \in \mathbb{N}\}\), where \(e_n = (0,...,1,0...)\) has 1 on the \(n\)-th index. Then \(Z = \{\sum_{i=1}^N c_ne_n \mid N \in \mathbb{N}, c_i \in \mathbb{C}\}\). Then it can be verified that \(Z\) is a linear subspace of \(l^p(\mathbb{C})\). Pick \(p=2\), then \(x = (1, 1/2, 1/3, ...) \in l^2\) but not in \(Z\), which means \(Z\) is not closed in\(l^2\). Thus, by Theorem1.1 we know that \((Z, ||\cdot||_p)\) is not a Banach space.

Definition: \(C_{00}\)

\(C_{00} = \{\{a_n\}, a_n \rightarrow 0 \sim n \rightarrow \infty\}\).

Proposition1.2:

\(C_{00}\) is dense in \(l^p\) but not closed.

proof:

Let \(a = (a_1, a_2, ...) \in l^p\). Take \(b_N = (a_1, a_2,..., a_N, 0, 0, ...) \in C_{00}\). But \(||a - b_N||_{l^p} = \sum_{y=n+1}^{\infty}||a_{N+1}||^p \rightarrow 0 \sim N \rightarrow \infty\). \(\Box\)

Remark: If \(X\) is an infinite-dimensional Banach space, then any of its Hamel basis is uncountable. This means that \(C_{00}\) is not a Banach space in any norm.