What is Equivalent norms? — Mathematics

Definition 5.3 University Mathematics — Year 2 · Chapter 5 — Normed Vector Spaces

Two norms $N_1, N_2$ on $E$ are equivalent when there are constants $c, C > 0$ with

c\,N_1 \leq N_2 \leq C\, N_1 .

Equivalent norms have the same open sets, the same convergent and Cauchy sequences, the same compact and complete subsets: the same analysis.

The unit balls of the three classical norms of ℝ2, nested as the inequalities of dictate: smaller ball, larger norm. Roundness matters: the flat sides of the diamond and the square are exactly the failures of strict convexity exploited in ’s weekend problem and in this chapter’s (question 4). — The unit balls of the three classical norms of $\R^2$ , nested as the inequalities of Example 5.5 dictate: smaller ball, larger norm. Roundness matters: the flat sides of the diamond and the square are exactly the failures of strict convexity exploited in Chapter 8’s weekend problem and in this chapter’s (question 4).

Examples

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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