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A homework problem:

Let $H$ be a Hilbert space. Let $T:H\rightarrow H$ be a symmetric linear map ($\langle Tx,y\rangle=\langle x,Ty\rangle$).

Show that $S$ is bounded.

My attempt: I'd like to use the closed graph theorem. I take $(x_n)\subset H$ and assume $x_n \rightarrow x$ and $Tx_n\rightarrow y$. I'd like to show $Tx=y$. So I calculate: $\|Tx_n-Tx\|^2=\|T(x_n-x)\|^2=|\langle T(x_n-x), T(x_n-x)\rangle|=$ $|\langle x_n-x, T(T(x_n-x))\rangle|\leq \|x_n-x\|\cdot \|T(T(x_n-x))\|$.

So, it's enough to show that $\|T(T(x_n-x))\|$ is bounded. The fact that $T(x_n)$ converges tells me that $\|T(x_n-x)\|$ is bounded, but I don't know what about $\|T(T(x_n-x))\|$.

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You must prove that if $x_n\to 0$ and $Tx_n\to y$ then $y=0$. –  Alireza Nov 29 at 20:10

1 Answer 1

up vote 2 down vote accepted

Assume $x=0$ (which is possible by linearity, working if necessary with $x'_n:=x_n-x$, so $Tx'_n\to Tx_n-Tx=y-Tx$), and write $$\langle y,y\rangle=\lim_{n\to +\infty}\langle Tx_n,y\rangle\underset{\color{red}{\mbox{sym}}}{=}\lim_{n\to +\infty}\langle x_n,Ty\rangle,$$ and conclude using the fact that $x_n\to 0$.

(so actually, in $\lVert y-Tx_n\rVert$, we just need to take the limit at one side of the inner product, not both).

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How does the map being symmetric tell that the map is bounded . I don't get it. In your answer what is $y$, is it $0$ ? –  Theorem Dec 10 '12 at 21:12
    
@Theorem I've added details. –  Davide Giraudo Dec 10 '12 at 21:15

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