Linear Algebra and Its Applications · Problem 7.2.19

Computations with Matrices — Gilbert Strang

Mathematics · vector-norms

Problem

Prove that

\|x\|_\infty \leq \|x\| \leq \|x\|_1

. Show from the Schwarz inequality that the ratios

\|x\|/\|x\|_\infty

and

\|x\|_1/\|x\|

are never larger than

\sqrt{n}

. Which vector

(x_1,\ldots,x_n)

gives ratios equal to

\sqrt{n}