Weak convergence for linear operator in Hilbert Space

Question

Assume $T : X \to Y$ be a linear operator for $X,Y$ Hilbert spaces. Moreover, let $<\,\cdot\, , \,\cdot\,>_{X}$ and $<\,\cdot\, , \,\cdot\,>_{Y}$ be inner dot product of $X$ and $Y$ respectively and assume $x_{n} \to x$ in $X$ weakly and $Tx_{n} \to y$ weakly in $Y$. I want to show that $Tx = y$. I can only obtain that $$<T(x_{n}-x),\eta>_{Y}\to <y-Tx,\eta>_{Y}$$ for any $\eta \in Y$ as $n\to\infty$ but I do not know how to proceed.

Answer 1

Not true if $T$ is just a linear map. I suppose you wanted to say that $T$ is a bounded linear map.

You can prove this easily using the adjoint $T^{*}$. We have $ \langle Tx, z \rangle =\langle x, T^{*}z \rangle =\lim \langle x_n, T^{*}z \rangle =\lim \langle Tx_n, z \rangle =\langle y, z \rangle$ for all $z$. Hence $Tx=y$.