is the adjoint of a linear operator also linear?

Question

How would you go about proving whether or not the adjoint of a linear operator $H$ (in a vector space) is also linear? I have

$$H: X \to Y \text{ and } H^*: X \to Y$$

such that

$$\langle H(x),y\rangle = \langle x,H^*(y)\rangle$$

I'm not sure where to start, any hints would be great and I can take it from there.

Answer 1

Hint for additivity: Show that $\langle x, H^*(y) + H^*(y') - H^*(y+y') \rangle = 0$ for all $x \in X$ and $y,y' \in Y$.