The Baire Category Theorem

Closed Graph Theorem

Let $X$ and $Y$ be Banach spaces and $T$ a linear map of $X\to Y$. Then $T$ is bounded if and only if the graph of $T$ is closed.

Helinger-Toeplitz Theorem

Let $A$ be an everywhere defined linear operator on a Hilbert space \newcommand{\H}{\mathcal{H}}\H such that $⟨x,Ay⟩=⟨Ax,y⟩$ holds for all x,y\in\H. Then $A$ is bounded.

Proof By the closed graph theorem, it suffices to show $\Gamma (A)$ is closed in \H\times\H. Suppose $\{(x_n,A{x}_n){\}}_{n=1}^{\infty }$ is a sequence in $\Gamma (A)$ that converges to $(x,y)$. Then $x_n\to x$ and $A{x}_n\to y$. By the continuity of the inner product, we have $⟨A(x_n-x),A(x_n-x)⟩=⟨A^2(x_n-x),x_n-x⟩\to 0.$Therefore by triangle inequality, we have $\Vert y-Ax\Vert \le \Vert y-A{x}_n\Vert +\Vert A{x}_n-Ax\Vert \to 0.$This shows that $y=Ax$, hence $(x,y)\in \Gamma (A)$, which implies that $\Gamma (A)$ is closed. $\square$