text: Q:
  
  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.
  
  A:
  
  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.
  
  