Let $V$ be an inner product (not necessarily finite dimensional). Let $\beta$ and $\gamma$ be fixed vectors.Then does there exist an adjoint of the linear operator of $T$ defined as $T(\alpha)=(\alpha|\beta)(\gamma)$ and if so then what will be the adjoint of $T$.
I am stuck in this question for a very long time. If $V$ is finite dimensional then adjoint of $T$ exists. If $V$ is not finite dimensional then how do I show that adjoint exists? I think if it exists then I will be able to show what it is.