I am trying to figure out what could be the norm associated with the product of two normed vector spaces.

I know we can define several norms on it, for example, if we have $(X, \|·\|)$ and $(Y, \|·\|)$ two normed vector spaces we can define the following norm on $X \times Y $ :

$$ \|(x, y)\|=(\|x\|^p+\|y\|^p)^{1/p}$$

So... Are all these norms equivalent in $X \times Y $ ? Is there a standard norm that can be defined $X \times Y $ ?

Note that $X$ and $ Y$ can be infinite dimensional.


You can choose a norm in $X \times Y$ as: $$ \|(x,y)\| = \varphi(\|x\|, \|y\|) $$ where $\varphi$ is arbitrary norm in $\mathbb R^2$.

Now consider two different norms $\varphi$ and $\psi$ in $\mathbb R^2$. They are obviously equivalent in $\mathbb R^2$, so there are some positive constants $m$ and $M$, such that (for $a \neq 0 \vee b \neq 0$): $$ m \le \frac{\varphi(a, b)}{\psi(a, b)} \le M $$

Now this is obviously also true for the corresponding norms in $X \times Y$ (for $x \neq 0 \vee y \neq 0$): $$ m \le \frac{\varphi(\|x\|, \|y\|)}{\psi(\|x\|, \|y\|)} \le M $$ Which means that they are equivalent.

Note that if you try to use function $\varphi$ which is not a norm in $\mathbb R^2$, then you'll get a function in $X \times Y$, which is not a norm too.

P.S. This is true for a product of any finite number of normed spaces.

  • $\begingroup$ Thanks, that solved my question. $\endgroup$
    – D1X
    Dec 19 '15 at 21:58

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