Normed Linear Spaces
Table of Contents

Normed Linear Spaces

Recall from the Linear Spaces page that a linear space over $\mathbb{R}$ (or $\mathbb{C}$) is a set $X$ with a binary operation $+$ defined for elements in $X$ and scalar multiplication defined for numbers in $\mathbb{R}$ (or $\mathbb{C}$) with elements in $X$ that satisfy ten properties (see the aforementioned page).

We said a subset $Y \subseteq X$ is a linear subspace of $X$ if $Y$ with the same binary operation $+$ and scalar multiplication $\cdot$ restricted to $Y$ is itself a linear space.

We are about to define a special type of linear space called a normed linear space. We will first need to define what a norm on a linear space is.

Definition: Let $X$ be a linear space over $\mathbb{R}$ (or $\mathbb{C}$). A Norm on $X$ is a nonnegative real-valued function on $X$, commonly denoted $\| \cdot \|$ such that:
1) $\| x \| = 0$ if and only if $x = 0$.
2) $\| \alpha x \| = | \alpha | \| x \|$ for all $\alpha \in \mathbb{R}$ (or $\mathbb{C}$) and for all $x \in X$.
3) $\| x + y \| \leq \| x \| + \| y \|$ for all $x, y \in X$.
If only properties (2) and (3) from above hold, then $\| \cdot \|$ is instead called a Seminorm.
Definition: A Normed Linear Space is a pair $(X, \| \cdot \|)$ where $X$ is a linear space and $\| \cdot \| : X \to [0, \infty)$ is a norm on $X$.

The terms "normed linear space", "normed vector space", and "normed space" can be used interchangeably.

When we have a normed linear space $(X, \| \cdot \|)$, we can quite naturally obtain a metric space by defining a metric $d$ on $X$ in terms of the given norm $\| \cdot \|$.

Theorem 1: If $X$ is a normed linear space over $\mathbb{R}$ (or $\mathbb{C}$) with norm $\| \cdot \|$ and if $d : X \times X \to [0, \infty)$ is defined for all $x, y \in X$ by $d(x, y) = \| x - y \|$ then $(X, d)$ is a metric space.
  • Proof: We show that $d$ is a metric on $X$.
  • Let $x, y \in X$. Then:
(1)
\begin{align} \quad d(x, y) = \| x - y \| = \| -1(y - x) \| = |-1| \| y - x \| = 1 \| y - x \| = \| y - x \| = d(y, x) \end{align}
  • Let $x, y \in X$ and suppose that $d(x, y) = 0$. Then:
(2)
\begin{align} \quad 0 = d(x, y) = \| x - y \| \quad \Leftrightarrow \quad x - y = 0 \quad \Leftrightarrow \quad x = y \end{align}
  • Lastly, let $x, y, z \in X$. Then:
(3)
\begin{align} \quad d(x, z) = \| x - z \| = \| (x - y) + (y - z) \| \leq \| x - y \| + \| y - z \| = d(x, y) + d(y, z) \end{align}
  • Hence $d$ is a metric on $X$ and $(X, d)$ is a metric space. $\blacksquare$

We give a special name to the metric space that can be obtained by a normed linear space.

Definition: If $X$ is a normed linear space over $\mathbb{R}$ (or $\mathbb{C})$ with norm $\| \cdot \|$ then the Metric Space of $X$ Induced by $\| \cdot \|$ is the metric space $(X, d)$ where $d : X \times X \to [0, \infty)$ is defined for all $x, y \in X$ by $d(x, y) = \| x - y \|$.
Definition: If $X$ is a normed linear space over $\mathbb{R}$ (or $\mathbb{C})$ with norm $\| \cdot \|$ then $X$ is said to be Complete if the metric space of $X$ induced by $\| \cdot \|$ is a complete metric space.

Recall that a metric space $(X, d)$ is said to be complete if every Cauchy sequence in $X$ converges in $X$.

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