Show that this operator is linear

Question

Let $\Bbb H$ is a Hilbetr space and $T:\Bbb H\to\Bbb H$ be a operator such that $$<x,Ty>=<Tx,y>$$ $\forall x,y\in\Bbb H.$ I want to show that $T$ is linear and bounded. If I can show that $T$ is linear, then I can use "Closed Graph Theorem" to show the boundedness. But I have no idea about showing the linearity. Useful hint would be enough. Thank you.

Answer 1

Hint

Note that

$$<T(x_1+x_2),y>=<x_1+x_2,Ty>=<x_1,Ty>+<x_2,Ty>\\=<Tx_1,y>+<Tx_2,y>=<Tx_1+Tx_2,y>, \forall y \in \mathbb{H},$$ from where $T(x_1+x_2)=T(x_1)+T(x_2).$

Proceed in the same way with product by scalars.