If every $x\in X$ is uniquely $x=y+z$ then $\|z\|+\|y\|\leq C\|x\|$

functional-analysis

Solution 1:

Let $W:= Y \oplus Z$, which is a Banach space when endowed with the norm $\|(y,z)\|= \|y\|+\|z\|$.

Define a map from $W$ to $X$ by sending $(y,z) \mapsto y+z$. Note that this map is injective and surjective by the given conditions. Moreover this map is bounded because clearly $\|y+z\|\leq \|y\|+\|z\|$. By the open mapping theorem, the inverse map is bounded as well, so that $\|y\|+\|z\|\leq C\|x\|$.

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