What is norm? — Mathematics

Definition 5.1 University Mathematics — Year 2 · Chapter 5 — Normed Vector Spaces

A norm on $E$ is a map $\norm{\,\cdot\,} \colon E \to \R_+$ with, for all $x, y \in E$ , $\lambda \in K$ :

\norm x = 0 \iff x = 0, \qquad \norm{\lambda x} = \abs\lambda\,\norm x, \qquad \norm{x + y} \leq \norm x + \norm y .

Then $d(x, y) = \norm{x - y}$ is a distance, and all of Chapter 4 applies. The reverse triangle inequality $\bigl|\norm x - \norm y\bigr| \leq \norm{x - y}$ makes the norm itself $1$ -Lipschitz; addition and scalar multiplication are continuous (estimates $\norm{(x + y) - (x' + y')} \leq \norm{x - x'} + \norm{y - y'}$ , etc.).

Examples

Example 5.2

On $K^n$ :

\norm{x}_1 = \sum_i \abs{x_i}, \qquad \norm{x}_2 = \Bigl(\sum_i \abs{x_i}^2\Bigr)^{1/2}, \qquad \norm{x}_\infty = \max_i \abs{x_i}

( $\norm\cdot_2$ is a norm by Cauchy–Schwarz, Year 1 volume). On $C(\intcc{a}{b})$ :

\norm f_\infty = \sup \abs f, \qquad \norm f_1 = \int_a^b \abs f, \qquad \norm f_2 = \Bigl(\int_a^b \abs f^2\Bigr)^{1/2},

the last two being norms thanks to strict positivity of the integral and integral Cauchy–Schwarz (Year 1 volume). On matrices: any norm on $\mathcal M_n(K) \simeq K^{n^2}$ ; the operator norms below are the structurally important ones.

Example 5.4 (Non-equivalence in infinite dimension)

On $C(\intcc{0}{1})$ : $\norm f_1 \leq \norm f_\infty$ always, but no reverse bound holds: $f_n(x) = x^n$ has $\norm{f_n}_\infty = 1$ and $\norm{f_n}_1 = \frac{1}{n+1} \to 0$ . So $f_n \to 0$ for $\norm\cdot_1$ but not for $\norm\cdot_\infty$ : the two norms disagree about convergence itself.

Example 5.5 (Explicit constants in dimension $n$ )

On $K^n$ the three classical norms are equivalent with sharp constants:

\norm x_\infty \leq \norm x_2 \leq \norm x_1 \leq \sqrt n\,\norm x_2 \leq n\,\norm x_\infty ,

the middle bound $\norm x_1 \leq \sqrt n\norm x_2$ coming from Cauchy–Schwarz against the all-ones vector. Extremal vectors: $e_1$ makes the first two inequalities equalities, $(1, 1, \dots, 1)$ the last two. The dimension $n$ sits visibly in the constants — the quantitative seed of the failure in infinite dimension: as $n \to \infty$ no uniform constant survives, which is exactly what Example 5.4 exhibits on function spaces.

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