Appendix: Functional Analysis

If $X, Y\in \mathcal{B}$ and $T:X\to Y$ is a bijective continuous operator, then $T^{-1}$ is continuous and thus $T$ is a homeomorphism.

If $X, Y\in \mathcal{B}$ and $T:X\to Y$ is a surjective continuous operator, then $T$ is an open map.

If $X, Y\in \mathcal{B}$ and $T \in L(X, Y)$ is a closed linear operator, i.e. the graph $\Gamma(T) \subseteq X\times Y$ is closed, then $T$ is bounded.

Let $X, Y\in \mathcal{B}$ and $\left\{{ T_{\alpha}}\right\} \subseteq L(X, Y)$ be a family of uniformly pointwise bounded operators, so for all points $x$ there exists a constant $C_x$ such that ${\left\lVert {T_{\alpha}x} \right\rVert} \leq C_x$ for all $\alpha$. Then there exists a constant bound that is uniform in $x$, i.e. a $C$ such that ${\left\lVert {T_{\alpha}x} \right\rVert}\leq C$ for all $x$.

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