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:
- Let $x, y \in X$ and suppose that $d(x, y) = 0$. Then:
- Lastly, let $x, y, z \in X$. Then:
- 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 $X$.
Proposition 1 (Reverse Triangle Inequality): Let $(X, \| \cdot \|_X)$ be a normed linear space. Then for all $x, y \in X$ we have that $| \| x \|_X - \| y \|_X | \leq \| x - y \|_X$. |
- Proof: We have that:
- Therefore:
- Similarly we have that:
- Therefore:
- And so:
- Hence:
Proposition 2: Let $(X, \| \cdot \|_X)$ be a normed linear space. Then the function $\| \cdot \|_X : X \to \mathbb{R}$ is a continuous function on $X$. |
- Proof: Let $x_0 \in X$ and let $\epsilon > 0$ be given. Let $\delta = \epsilon$. Then if $\| x - x_0 \|_X < \delta = \epsilon$ we have that:
- So $\| \cdot \|_X : X \to \mathbb{R}$ is continuous at $x_0$. Since $x_0$ was arbitrary, $\| \cdot \|_X : X \to \mathbb{R}$ is continuous on $X$. $\blacksquare$
Proposition 3: Let $(X, \| \cdot \|_X)$ be a normed linear space. If $(x_n)$ converges to $x \in X$ then $\| x_n \|_X$ converges to $\| x \|_X$. |
- Proof: Suppose that $(x_n)$ converges to $x$. Then for all $\epsilon > 0$ there exists an $n \in \mathbb{N}$ such that if $n \geq N$ then $\| x_n - x \|_X < \epsilon$. For all $n \in \mathbb{N}$ we have that:
- So if $n \geq N$ we have that $|\| x_n \|_X - \| x \|_X| < \epsilon$ and so $(\| x_n \|_X)$ converges to $\| x \|_X$. $\blacksquare$