# Equivalence of vector norms

#### Mollier

Hi.

problem:
For each of the following, verify the inequality and give an example of a nonzero vector or matrix for which equality is achieved. In this problem x is an m-vector and A is an $$\displaystyle m \times n$$ matrix.

(a) $$\displaystyle ||x||_{\infty} \leq ||x||_2$$,

(b) $$\displaystyle ||x||_2 \leq \sqrt{m} ||x||_{\infty}$$,

(c) $$\displaystyle ||A||_{\infty} \leq \sqrt{n} ||A||_2$$,

(d) $$\displaystyle ||A||_2 \leq \sqrt{m} ||A||_{\infty}$$.

attempt:

(a) I rewrite the inequality so it is easier for me to read:
$$\displaystyle (max_i |x_i|)^2 \leq (x^2_1+\cdots +x^2_m)$$
Let $$\displaystyle x_k$$ be the largest component of $$\displaystyle x$$. Then $$\displaystyle (max_i|x_i|)^2=x^2_k$$.
If the remaining components are zero, the inequality becomes an equality. If one or more of the remaining components are nonzero, it becomes an inequality.
Example of vector: $$\displaystyle x=[x_k,0,\cdots,0]$$.

I'm sure there are better ways to verify this.

(b) Again, I rewrite:

$$\displaystyle x^2_1+\cdots+x^2_m \leq m(max_i|x_i|)^2$$.
Let $$\displaystyle x_k$$ be the largest component of $$\displaystyle x$$.
Since $$\displaystyle mx^2_k=x^2_k+\cdots +x^2_k$$ (m-times) and since $$\displaystyle x_i$$ for$$\displaystyle i\neq k$$ are smaller than $$\displaystyle x_k$$, the inequality holds.
It's an equality for all vectors $$\displaystyle x$$ with $$\displaystyle m=1$$

I am not at all sure what to do with (c) and (d).
Any hints are appreciated.

Is $$\displaystyle ||A||_p$$ the regular p norm when A is regarded as a vector in $$\displaystyle \mathbb{R}^{nm}$$?

#### Mollier

Is $$\displaystyle ||A||_p$$ the regular p norm when A is regarded as a vector in $$\displaystyle \mathbb{R}^{nm}$$?
$$\displaystyle ||A||_p=\left( \sum^m_{i=1} \sum^n_{j=1} |a_{ij}|^p \right)^{1/p}$$

Is this what you mean?

Yes. A matrix is a block of nm numbers, so it can be identified with a vector in R^{nm}, and its norm is the same as the norm of this vector, so this makes (a) stronger than (c) I believe. For (d), consider an n by 1 matrix all of whose entries are C>0. Then (d) asserts $$\displaystyle nmC^2 \leq mC^2$$ which is false if n>1, so perhaps there was a typo? If m is replaced by nm then (d) follows from (b) in the same way that (c) follows from (a).
If I look at the matrix A as a $$\displaystyle mn$$-dimensional vector and then calculate the, say 2-norm, I am calculating the Forbenius norm. $$\displaystyle ||A||_2$$ is the matrix norm induced by vector norms, and so is not the same as the Forbenius norm.