Equivalent Norms¶

Definition¶

Two norms $\|\cdot\|_p$ and $\|\cdot\|_q$ on a vector space $V$ (the subscripts are just labels for two arbitrary norms, not necessarily $p$ -norms) are termed equivalent if there exist constants $a, b > 0$ such that for every $\boldsymbol{v} \in V$ $V$ , we have:

a \|\boldsymbol{v}\|_p \leq \|\boldsymbol{v}\|_q \leq b \|\boldsymbol{v}\|_p.

For example, the Euclidean Norm and maximum norm on $\mathbb{R}^n$ satisfy $\|\boldsymbol{v}\|_\infty\le\|\boldsymbol{v}\|_2\le\sqrt n\,\|\boldsymbol{v}\|_\infty$ $n ∥ v ∥_{\infty}$ . These constants show precisely how their lengths are comparable.

Interpretation¶

Equivalent norms are different ways of measuring length that agree up to fixed scaling factors: a vector that is short in one norm is short in the other, and the constants $a$ and $b$ say by how much the two lengths can differ. Geometrically, the unit circle of one norm fits between two scaled copies of the unit circle of the other.

In the plane, the unit circle of the Euclidean norm drawn between the unit square of the maximum norm and a smaller square scaled by 1 over the square root of 2.

On $\mathbb{R}^2$ , the illustrated norms satisfy $\|\boldsymbol{x}\|_\infty\le\|\boldsymbol{x}\|_2\le\sqrt{2}\,\|\boldsymbol{x}\|_\infty$ $2 ∥ x ∥_{\infty}$ , so they are equivalent. In the figure, the Euclidean unit circle lies inside the square $\|\boldsymbol{x}\|_\infty=1$ $1$ , which is the left inequality, and outside the dashed square $\|\boldsymbol{x}\|_\infty=\tfrac{1}{\sqrt{2}}$ $\frac{1}{2}$ , which is the right inequality.

Pick $p$ and $q$ : the $q$ -circles of radius $a$ and $b$ squeeze the $p$ -unit circle between them, so $a\|\boldsymbol v\|_p\le\|\boldsymbol v\|_q\le b\|\boldsymbol v\|_p$ $b ∥ v ∥_{p}$ .

The constants must not depend on $\boldsymbol{v}$ . This uniformity is what makes equivalent norms interchangeable in analysis: the metrics $\|\boldsymbol{v}-\boldsymbol{w}\|$ they define are strongly equivalent, so they have the same convergent sequences and the same Cauchy sequences. On $\mathbb{R}^n$ , where all norms are equivalent, the choice between the Euclidean, taxicab and maximum norm is a matter of convenience; the constants usually depend on $n$ , as in $\|\boldsymbol{v}\|_2\le\sqrt{n}\,\|\boldsymbol{v}\|_\infty$ $n ∥ v ∥_{\infty}$ .

A common slip is to take equivalence of all norms for granted in every vector space. It is a finite-dimensional fact. For continuous functions on $[0,1]$ , the uniform norm and $\int_0^1|f(x)|\,\mathrm{d}x$ are both norms, but $f_n(x)=x^n$ has uniform norm $1$ while the integral $\tfrac{1}{n+1}$ tends to $0$ , so no constant $a>0$ gives $a\|f_n\|_\infty\le\int_0^1|f_n(x)|\,\mathrm{d}x$ for all $n$ .

Key Results¶

Any two norms $\|\cdot\|_p$ and $\|\cdot\|_q$ on a finite-dimensional vector space $V$ are equivalent.

Examples¶

The Euclidean norm and the taxicab norm are equivalent.

Prerequisites¶

Sources¶

Math III: Analysis · Lecture 4

Aliases: norm equivalence Symbols: element of, infinity, integral, real numbers, square root

Last updated: 20 October 2026 03:02