What is norm?

Example 5.2

On $K^n$:

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

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:

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.