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Problem: Suppose $X$ is a Hilbert space. $A,B$ are two linear operators from $X$ to $X$ such that $(Ax , y)=(x , By)$ holds for every $x , y \in X$ ((.) means inner product). Prove that $A,B$ are both bounded.

This is a midterm test problem from my school. I have some thoughts but I am so confused that all theorems used to solve this kind of problem needs a bounded operator so I was trying to prove it by contradiction but I have no idea how to use the unbounded and given Hilbert space conditions. I feel like it's some kind of symmetric related conclusion but still can't figure it out.

Felix
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  • Does this answer your question ?https://math.stackexchange.com/questions/4857413/if-langle-lx-y-rangle-langle-x-ry-rangle-then-l-is-bounded/4857465#4857465 – Ryszard Szwarc May 13 '24 at 08:59

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This is an easy consequence of the Closed Graph Theorem.

Let $x_n \to u$ and $Ax_n \to v$. Then $(Ax_n,y)=(x_n, By)$ so $(v,y)=(u,By)$. But $(u,By)=(Au,y)$ so we get $(v,y)=(Au,y)$. This is true for all $y$, so $v=Au$ proving that $A$ is bounded. Similarly $B$ is bounded.

Kavi Rama Murthy
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Hint: First show that $A$ and $B$ are closed operators (i.e. for every sequence $x_n \to x$, if $Ax_n \to y$ then $Tx =y$). Now apply the closed graph theorem.

approximate-identity
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