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Let $(X,\|\cdot\|_X) $ be an infinite dimensional Banach space. Suppose that you can write $X=V\oplus W$. Write $x=v+w$ and define $(V\oplus W,\|x\|_\oplus :=\|v\|_X+\|w\|_X)$. Show that $\|\cdot\|_\oplus$ is equivalent to $\|\cdot\|_X$.

One of the inequalities is easy: $$ \|x\|_X=\|v+w\|_X\leq \|v\|_X+\|w\|_X=\|x\|_\oplus$$

How about $C\|x\|_\oplus \leq\|x\|_X$ ?

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Use the Bounded Inverse Theorem. – David Mitra Jan 13 '13 at 22:22
up vote 2 down vote accepted

Hint: You show that $(X,\|.\|_\oplus)$ is a banach space and apply the Open Map theorem to identity $I:(X,\|.\|_\oplus)\to(X,\|.\|_X)$ (You already proved that $I$ is continuous)

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Hint: What you've done so far shows that the bijective, linear map $T:V\oplus W\rightarrow X$ defined by $x\oplus y\mapsto x+y$ is continuous. To obtain the reverse inequality, appeal to the Bounded Inverse Theorem.

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