I am trying to prove the existence of adjoints of bounded linear operators on Hilbert spaces:
If $H,H'$ are Hilbert spaces and $u \in B(H,H')$ then there exists a unique $u^\ast \in B(H',H)$ such that for all $h \in H, h' \in H'$:
$$ \langle u(h), h'\rangle = \langle h, u^\ast (h') \rangle$$
Here is what I have so far:
Since $u$ is bounded and linear, $h \mapsto \langle u(h), h'\rangle$ is in $H^\ast$. Therefore by the Riesz representation theorem there exists $h_0 \in H$ such that
$$ \langle u(h), h'\rangle = \langle h, h_0\rangle$$
$$ u^\ast (h') = h_0$$
and claim that $u^\ast$ is linear and continuous.
Linearity: Consider $u^\ast (\lambda h' + \mu h'')$. By how $u^\ast$ was defined, for all $h \in H$:
$$ \langle h, u^\ast (\lambda h' + \mu h'') \rangle = \langle u(h), \lambda h' + \mu h''\rangle$$
Now I get stuck because the right argument is conjugate linear, not linear.
How can I show that $u^\ast$ is linear?