**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.