$\newcommand{R}{\mathbb{R}} \newcommand{C}{\mathbb{C}} \newcommand{defarrow}{\quad \stackrel{\text{def}}{\Longleftrightarrow} \quad}$

# Normed spaces

## Definition

A normed space is a vector space $X$ endowed with a function $X \to [0,\infty), \qquad x \mapsto \| x \|,$ called the norm on $X$, which satisfies: \begin{align*} &\mathrm{(i)} \quad &\|\lambda x\| &= |\lambda|\, \|x\|, \qquad&\text{(positive homogeneity)}&\\ &\mathrm{(ii)} \qquad& \|x + y\| &\leq \|x\| + \|y\|, \qquad&\text{(triangle inequality)}&\\ &\mathrm{(iii)} \quad\; &\|x\| = 0 \quad&\text{if and only if }\quad x = 0, \qquad&\text{(positive definiteness)}&\\ \end{align*} for all scalars $\lambda$ and all elements $x, y \in X$. A vector space may allow for many different norms, but not all vector spaces are normable.1)

Ex.
• The vector space $\R^n$ with the usual addition and scalar multiplication allows for several norms, for example:

$\quad$ the Euclidean norm $\|(x_1,\ldots, x_n)\|_{l_2} = \big( x_1^2 + \ldots + x_n^2 \big)^{1/2}$ $\quad$ the maximum norm $\|(x_1,\ldots, x_n)\|_{l_\infty} = \mathrm{max} \{ |x_1|, \ldots, |x_n| \},$ $\quad$ and the summation norm $\|(x_1,\ldots, x_n)\|_{l_1} = |x_1| + \ldots + |x_n|.$ $\quad$ These are all special cases of the (finite-dimensional) $l_p$-norm $\|(x_1,\ldots, x_n)\|_{l_p} = \big( \sum_{j = 1}^n |x_j|^p \big)^{1/p}$, $1 \leq p \leq \infty$.

Proof (of the example)

Proof (of the example)

For both $\|\cdot\|_{l_2}$, $\|\cdot\|_{l_\infty}$ and $\|\cdot\|_{l_1}$, it is clear that they are non-negative functions, and that $\| x \| = 0 \quad\Longleftrightarrow\quad x = (x_1, \ldots, x_n) = (0,\ldots,0).$ In addition, $\| \lambda x \|_{l_2} = \big( (\lambda x_1)^2 + \cdots + (\lambda x_n)^2 \big)^{1/2} = |\lambda| \big( x_1^2 + \cdots + x_n^2 \big)^{1/2} = |\lambda| \|x\|_{l_2},$ and similarly for $\|\cdot\|_{l_\infty}$ and $\|\cdot\|_{l_1}$.

The triangle inequality for $\|\cdot\|_{l_\infty}$ and $\|\cdot\|_{l_1}$ follows from that on $\R$:

\begin{align} \|x +y\|_{l_1} &= \sum_{j=1}^n |x_j + y_j| \leq \sum_{j=1}^n ( |x_j| + |y_j| ) = \sum_{j=1}^n |x_j| + \sum_{j=1}^n |y_j| = \|x\|_{l_1} + \|y\|_{l_1}, \end{align}

\begin{align} \|x +y\|_{l_\infty} &= \mathrm{max} \{ |x_1 + y_1|, \ldots, |x_n + y_n| \}\\ &\leq \mathrm{max} \{ |x_1| + |y_1|, \ldots, |x_n| + |y_n| \}\\ &\leq \mathrm{max} \{ |x_1|, \ldots, |x_n| \} + \mathrm{max} \{ |y_1|, \ldots, |y_n| \}\\ &= \|x\|_{l_\infty} + \|y\|_{l_\infty}. \end{align}

The triangle inequality for $\|\cdot\|_{l_2}$ is a consequence of the Cauchy–Schwarz inequality, which we will prove later.

Ex.
• The space of real- (or complex-) valued bounded and continuous functions on an interval (open or closed), $BC(I,\R)$, becomes a normed vector space when endowed with the supremum norm2) $\|f\|_\infty = \sup_{x \in I} |f(x)|.$ If $I = [a,b]$ it follows from the extreme value theorem that $BC([a,b],\R) = C([a,b],\R)$ (as sets and linear spaces) and $\|f\|_{\infty} = \sup_{x \in [a,b]} |f(x)| = \max_{x \in [a,b]} |f(x)|.$ If $I = (a,b)$ is either infinite or does not contain its end points, then $BC((a,b),\R) \subsetneq C((a,b),\R)$. An example of this strict inclusion is the function $x \mapsto 1/x$ on (0,1). It is continuous, but $[x \mapsto 1/x] \not\in BC((0,1),\R),$ since $\sup_{x \in(0,1)} |1/x| = \infty.$

## Equivalence of norms

Two norms $\|\cdot\|_1$ and $\|\cdot\|_2$ on a vector space $X$ are said to be equivalent if there exists a number $c \in \R$ such that $c^{-1} \|x\|_1 \leq \|x\|_2 \leq c \|x\|_1 \qquad \text{ for all } \quad x \in X.$

Ex.
• The maximum and summation norms are equivalent on $\R^n$, since $\max\limits_{1 \leq j \leq n} |x_j| \leq \sum_{j=1}^n |x_j| \quad\text{ and }\quad \sum_{j=1}^n |x_j| \leq n \max\limits_{1 \leq j \leq n} |x_j|.$ Hence $n^{-1} \|x\|_{l_\infty} \leq \|x\|_{l_1} \leq n \|x\|_{l_\infty} \qquad\text{ for }\quad x = (x_1,\ldots, x_n).$
• One can show that, on a finite-dimensional vector space, any two norms are equivalent. In particular, any norm on $\R^n$ is equivalent to the Euclidean norm.
1)
Using the axiom of choice it is possible to assign a norm to any vector space, but this norm may not correspond to any natural structure of the space. For example, there is no norm such that $C^\infty(\R,\R)$, the set of infinitely differentiable real-valued functions on $\R$, is complete.
2)
The supremum of a set $A \subset \R$ is the smallest $M \in \R$ such that $a \leq M$ for all $a \in A$. If no such finite $M$ exists, then $\sup(A) = \infty$. One furthermore defines $\sup(\emptyset) = -\infty$. Thus, the supremum always exists. In a similar fashion, the infimimum of a set $A$ is the largest lower bound on the set; it can be defined as $\inf(A) = -\sup(-A)$, where $-A = \{ -a \in \mathbb R \colon a \in A \}$.