Normed vector spaces and Banach spaces

In the following let $X$ be a linear space (vector space) over the field $\mathbb{F} = {\mathbb{R}, \mathbb{C}}$.

Definition 1.1.

A seminorm on $X$ is a map $p: X \to \mathbb{R}_{+}$ such that

1. $p(\alpha x) = |\alpha|p(x)$ for all $\alpha \in \mathbb{F}, x \in X$
2. $p(x + y) \leq p(x) + p(y)$ for all $x, y \in X$
3. $p(x) = 0 \Rightarrow x = 0$

then $p$ is called a norm.
　Usually one writes $p(x) = \|x\|$, $p= \|\cdot\|$.
　The pair $(X, \|\cdot\|)$ is called a normed vector space.

Definition 1.2.

Let $(x_n)_{n \in \mathbb{N}}$ be a sequence in a normed vector space $X(X, \|\cdot\|)$. then

• $(x_n)$ converges to a limit $x \in X$ if $\forall \varepsilon > 0, \exists N \in \mathbb{N}$ such that for all $n \geq N$, $\|x_n - x\| < \varepsilon$.
• $(x_n)$ is a Cauchy sequence if $\forall \varepsilon > 0, \exists N \in \mathbb{N}$ such that for all $m, n \geq N$, $\|x_n - x_m\| < \varepsilon$.
• $(X, \|\cdot\|)$ is complete if every Cauchy sequence in $X$ converges.
• A complete normed vector space is called a Banach space.

Metric spaces

Definition

Given a set $M\neq \emptyset$, a metric (or distance) $d$ on $M$ is a map $d: M \times M \to \mathbb{R}$ such that

1. $d(x, y) \geq 0$ for all $x, y \in M$ and $d(x, y) = 0 \Leftrightarrow x = y$
2. $d(x, y) = d(y, x)$ for all $x, y \in M$
3. $d(x, y) \leq d(x, z) + d(z, y)$ for all $x, y, z \in M$

The par $(M, d)$ is called a metric space.
A sequence $(x_n)_{n \in \mathbb{N}}$ in a metric space $(M, d)$ $converges$ to $x \in M$ if $\forall \varepsilon > 0, \exists N \in \mathbb{N}$ such that for all $n \geq N$, $d(x_n, x) < \varepsilon$.