Any two norms on product space are equivalent

by Believer   Last Updated September 11, 2019 19:20 PM

Suppose I have two normed spaces $(X,||•||_x)$& $(Y,||•||_Y)$. Say, I construct a new space X×Y, then how can I show that any two norms on X × Y coming from$ ||•||_X $and$ ||•||_Y$ are equivalent. If X and Y are finite dimensional, then so is X × Y, so I am ok with this case. How to go with infinite dimensional case?

Tags : normed-spaces

Answers 1

If the way to obtain the norm on the product is by means of using a norm on $\mathbb R^2$, then you can do the following. Say $f,g$ are norms in $\mathbb R^2$ and you define $$ \|(x,y)\|_f=f(\|x\|_X,\|y\|_Y),\ \ \ \ \|(x,y)\|_g=g(\|x\|_X,\|y\|_Y). $$ As all norms in $\mathbb R^2$ are equivalent, there exist $\alpha,\beta>0$ with $\alpha f\leq g\leq \beta f$. Then $$ \alpha\|(x,y)\|_f=\alpha f(\|x\|_X,\|y\|_Y)\leq g(\|x\|_X,\|y\|_Y)=\|(x,y)\|_g\leq \beta f(\|x\|_X,\|y\|_Y)=\beta\|(x,y)\|_f. $$

Martin Argerami
Martin Argerami
September 11, 2019 19:17 PM

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