Let $E$ be a Banach space and $T:E\to E^*$ be a linear operator such that
(a) $\langle Tx,x\rangle\ge 0,\,\,\,\,\forall x\in E $, by applying closed graph theorem prove that $T$ is a bounded operator.
(b) $\langle Tx,y\rangle=\langle Ty,x\rangle,\,\,\,\,\forall x,y\in E $, Prove that $T$ is a bounded linear operator.
So far I did the following:
(a) If we can prove that the graph of $T$ is closed then by closed graph theorem $T$ is continuous as well as bounded. Let $\{x_n\}\in E$ be a sequence such that $x_n\to x$ and $Tx_n\to f$, then passing the limit to the inequality
$$\langle T(x_n-y),x_n-y\rangle\ge 0$$
we get
$$\langle f-Ty,x-y\rangle\ge 0,\,\,\,\,\,\,\forall y\in E$$
Therefore by letting $y=x+tz,t\in\mathbb{R},z\in E$ we get the result.
(b) Sorry to bother you that I am stuck. Any help would be highly appreciated. Thanks in advance.