# 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?

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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
September 11, 2019 19:17 PM

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