Q:

Determine the adjoint of an operator

Let $u\in\mathcal{H}_1$ and $v\in\mathcal{H}_2$ and $T\in\mathcal{B}(\mathcal{H}_1,\mathcal{H}_2)$ be a continuous linear operator on hilbert space. This operator is defined by $Tf:=\langle f,u\rangle_{\mathcal{H}_1}v$. I want to find  the adjoint $T^*$.
Let $x\in\mathcal{H}_2$, then $\langle Tf,x\rangle_{\mathcal{H}_2}=\langle \langle f,u\rangle_{\mathcal{H}_1}v,x\rangle_{\mathcal{H}_2}=\langle u,f\rangle_{\mathcal{H}_1}\langle x,v\rangle_{\mathcal{H}_2}$
All my attempts to connect the inner products to obtain a $T^*$ with $\langle x,T^*f\rangle_{\mathcal{H}_1}$ failed so far. 
I would appreciate a little push into the right direction.
EDIT: All hilbert space mentioned is real.

A:

You are looking for an operator $T^*$ satisfying $\langle f, T^* x\rangle_{\mathcal{H}_1}=\langle Tf, x\rangle_{\mathcal{H}_2}$
As you wrote (you basically solved it),
$$\langle Tf,x\rangle_{\mathcal{H}_2}=\langle \langle f,u\rangle_{\mathcal{H}_1}v,x\rangle_{\mathcal{H}_2}=\langle u,f\rangle_{\mathcal{H}_1}\langle x,v\rangle_{\mathcal{H}_2}=\langle \langle x,v\rangle_{\mathcal{H}_2}u,f\rangle_{\mathcal{H}_1}$$
So $T^*x = \langle x,v\rangle_{\mathcal{H}_2}u$

