Show that $T+S$ is firmly nonexpansive considering that $T$ and $S$ are firmly nonexpansive mappings from $\mathbb R^n$ to $\mathbb R^n$.
Definition: We say that $F$ is firmly nonexpansive if: $$\|F(x)-F(y)\|^2+\|(I-F)(x)-(I-F)(y)\|^2\le \|x-y\|^2$$
2026-05-14 20:10:33.1778789433
Showing that $T+S$ is firmly nonexpansive
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[copied from comment]
It is false. Let $T = S = I$. Then $T+S = 2I$ which once you plug in you see that is not firmly nonexpansive. Do you perhaps mean to consider $\frac12(T+S)$? In which case the desired conclusion follows from triangle inequality applied many times.