Fix $u,x \in V$ and define $T\in \mathcal{L}(V)$ by $$Tv = \langle v, u \rangle x $$
I am trying to find the adjoint operator, but not sure how to use the inner product properties to get $T*$. Here is what I have so far
Let $w_1, w_2 \in V$. Then $\langle w_1, T^*w_2 \rangle = \langle Tw_1, w_2 \rangle = \langle \langle w_1, u \rangle x, w_2 \rangle$ but not sure where to go from here.
Best Answer
By using what you have shown
$$\langle w_1,T^\ast w_2\rangle = \langle \langle w_1,u\rangle x,w_2\rangle \underbrace{=}_{\text{linearity 1st arg}} \langle w_1,u\rangle \langle x,w_2\rangle \underbrace{=}_{\text{linearity 2nd arg}} \langle w_1,\langle x,w_2\rangle u\rangle$$