Bounding the infinity norm using the l_2 norm

Question

Suppose $\Big(\sum_{k =1}^n x_k^2\Big)^{1/2} \leq \eta_2$.

$\eta_2$ is the bound on the l_2 norm of a matrix. I want to upper bound $\eta_\infty = \max_{i}|x_i|$ using $\eta_2$ the bound on the l_2 norm of the matrix.

Is there a standard approach to do this?

Answer 1

For each $i$, $x_i^{2} \leq \sum_{k=1}^{n} x_k^{2}$ so $|x_i| \leq \sqrt {\sum_{k=1}^{n} x_k^{2}} \leq \eta_2$. Taking sup over $i$ you get $\eta_{\infty} \leq \eta_2$.