Banach spaces, $\lVert\cdot\rVert_X$ and $\lVert\cdot\rVert_{X+X}$ are equivalent.

Question

Let $(X,\lVert\cdot\rVert_X)$ and $(Y,\lVert\cdot\rVert_Y)$ be Banach spaces. Then as I understand, $X+Y$ endowed with $$\lVert v\rVert_{X+Y}=\inf\limits_{a+b=v\\ a\in X\\b\in Y}\lVert a\rVert_X+\lVert b\rVert_Y$$ is also a Banach space. Now I want to show that if $X=Y$ than the two norms are equivalent. Taking $a = v, b = 0,$ I have $\lVert v\rVert_{X+X}\le \lVert v\rVert_X$ but how to show $\lVert v\rVert_X\le C\lVert v\rVert_{X+X}$?

Answer 1

In general your notations don't make sense. You cannot add elments of two Banach spaces in general. But if $X$ and $Y$ are subspaces of one big vector space then this question makes sense.

$\|v\|_X \leq \|a\|_X+\|b\|_X$ for all choices of $a$ and $b$ with $v=a+b$. Taking infimum over all choices of $a$ and $b$ we get $\|v\|_X \leq \|v\|_{X+X}$.