1.2 Convergence and completeness in normed spaces

Convergence in normed linear spaces is a key concept in functional analysis. It defines how sequences of elements approach a limit point, using the norm to measure distance between elements.

Uniqueness of limits and Cauchy sequences are crucial ideas in this context. These concepts help us understand completeness, which ensures that every Cauchy sequence converges within the space, leaving no "gaps."

Convergence in normed linear spaces

Convergence in normed linear spaces

• Defines convergence of a sequence $(x_n)$ in a normed linear space $(X, \|\cdot\|)$ as $\lim_{n \to \infty} \|x_n - x\| = 0$ for some $x \in X$
• Denotes convergence as $x_n \to x$ or $\lim_{n \to \infty} x_n = x$
• Illustrates convergence using the space of continuous functions $C[0,1]$ with the supremum norm $\|f\|_\infty = \sup_{x \in [0,1]} |f(x)|$
  • Defines a sequence of functions $(f_n)$ by $f_n(x) = x^n$ for $x \in [0,1]$
  • Shows $(f_n)$ converges to the function $f(x) = 0$ for $x \in [0,1)$ and $f(1) = 1$

Uniqueness of limits

• Assumes $(x_n)$ converges to both $x$ and $y$ in a normed space $(X, \|\cdot\|)$
• Applies the triangle inequality to obtain $\|x - y\| \leq \|x - x_n\| + \|x_n - y\|$
• Takes the limit as $n \to \infty$, yielding $\lim_{n \to \infty} \|x - x_n\| = 0$ and $\lim_{n \to \infty} \|x_n - y\| = 0$
• Concludes $\|x - y\| \leq 0 + 0 = 0$, implying $x = y$
• Proves the uniqueness of limits in normed spaces

Cauchy sequences and completeness

• Defines a Cauchy sequence $(x_n)$ in a normed space $(X, \|\cdot\|)$ as one where for every $\varepsilon > 0$, there exists $N \in \mathbb{N}$ such that $\|x_m - x_n\| < \varepsilon$ for all $m, n \geq N$
• Explains intuitively that the terms of a Cauchy sequence become arbitrarily close to each other as the sequence progresses
• Relates Cauchy sequences to completeness by stating a normed space $(X, \|\cdot\|)$ is complete if every Cauchy sequence in $X$ converges to an element of $X$
• Emphasizes completeness ensures there are no "gaps" in the space, and every Cauchy sequence has a limit within the space

Completeness via Cauchy criterion

• Outlines proving a normed space $(X, \|\cdot\|)$ is complete by showing every Cauchy sequence in $X$ converges to an element of $X$
• Demonstrates completeness of $(\mathbb{R}, |\cdot|)$ as an example
  1. Lets $(x_n)$ be a Cauchy sequence in $\mathbb{R}$
  2. Applies the Cauchy criterion to obtain for every $\varepsilon > 0$, there exists $N \in \mathbb{N}$ such that $|x_m - x_n| < \varepsilon$ for all $m, n \geq N$
  3. Uses the completeness of $\mathbb{R}$ as an ordered field to show $(x_n)$ converges to a real number $x$
• Concludes $(\mathbb{R}, |\cdot|)$ is complete