# Is there a default norm for the (finite) product of Normed Vector Spaces?

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.

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