Problem:

verify the following inequalities:

(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 $

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Pathetic attempt:(a) $\displaystyle (\max_{1\leq i \leq m}|x_i|)^2 \leq (x^2_1+x^2_2+...+x^2_m) $

So, if the maximum absolute entry in the vector x is at index position, say i=2, then $\displaystyle ||x||_2$ is at least that big..

(b)

$\displaystyle \frac{\sqrt{x^*x}}{\sqrt{m}} \leq ||x||_\infty = \max_{1 \leq i \leq m}|x_i|$

The left side looks like the quadratic mean. Let's just say that the quadratic mean is always less than or equal to its largest absolute entry

(c)

Ugh, this is where I really get confused..

$\displaystyle (||A||_\infty)^2 \leq n\lambda_{max} $

Where $\displaystyle \lambda_{max}$ is the larges eigenvalue of $\displaystyle A^*A$...

(d)

No idea..

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These norms are giving me an abnormally hard time.

I've done quite some reading on them, using several sources, but I just can't wrap my head around them...

Any suggestions are greatly appreciated! Thanks.