Equivalence of Norms in Fⁿ

• categories: Real Analysis, Theorem, Topology

Equivalence of Norms in ${\mathbb{F}}^n$ In a finite-dimensional vector space ${\mathbb{F}}^n$ (where $\mathbb{F}$ is either $\mathbb{R}$ or $\mathbb{C}$), any two norms are equivalent. This means that if $\Vert \cdot {\Vert }_a$ and $\Vert \cdot {\Vert }_b$ are any two norms on ${\mathbb{F}}^n$, then there exist positive constants $C_1$ and $C_2$ such that for all $x\in {\mathbb{F}}^n$:

$$C_1\Vert x{\Vert }_a\le \Vert x{\Vert }_b\le C_2\Vert x{\Vert }_a.$$

Implications

The equivalence of norms implies that:

• Topology: All norms on ${\mathbb{F}}^n$ induce the same topology. Therefore, convergence, continuity, and open or closed sets in ${\mathbb{F}}^n$ are independent of the specific norm chosen.
• Continuity of Linear Maps: Any linear map from ${\mathbb{F}}^n$ to another normed space is continuous regardless of the norm on ${\mathbb{F}}^n$, since the equivalence of norms provides boundedness.

Example: Common Norms on ${\mathbb{F}}^n$

1. Euclidean Norm: $\Vert x{\Vert }_2={\left({\sum }_{i=1}^n|x_i{|}^2\right)}^{1/2}$.
2. 1-Norm: $\Vert x{\Vert }_1={\sum }_{i=1}^n|x_i|$.
3. Infinity Norm: $\Vert x{\Vert }_{\infty }={\max }_{1\le i\le n}|x_i|$.

For example, in ${\mathbb{R}}^n$, we have the relationships:

$$\Vert x{\Vert }_{\infty }\le \Vert x{\Vert }_2\le \sqrt{n}\Vert x{\Vert }_{\infty }$$

and

$$\Vert x{\Vert }_2\le \Vert x{\Vert }_1\le \sqrt{n}\Vert x{\Vert }_2.$$

These inequalities confirm that the norms $\Vert \cdot {\Vert }_1$, $\Vert \cdot {\Vert }_2$, and $\Vert \cdot {\Vert }_{\infty }$ are equivalent in ${\mathbb{F}}^n$.

Proof Sketch of Norm Equivalence in Finite Dimensions

For any norm $\Vert \cdot \Vert$ on ${\mathbb{F}}^n$, consider the unit sphere with respect to one norm, say $\Vert \cdot {\Vert }_a$. Since this sphere is compact in finite dimensions, $\Vert \cdot {\Vert }_b$ achieves a maximum and minimum value on this set. These values give the constants $C_1$ and $C_2$ that bound $\Vert \cdot {\Vert }_b$ in terms of $\Vert \cdot {\Vert }_a$, establishing equivalence.