TITLE: How to find adjoint of linear operator T on inner product space V
QUESTION [5 upvotes]: Let $V$ be an inner product space and $T$ a linear operator with $T(\alpha) = (\alpha,\beta)\gamma$ for fixed elements $\beta,\gamma \in V$.
I now that $T$ is linear operator. How we can show that adjoint of $T$ ($T^*$) exist and what is it?

REPLY [4 votes]: By the very definition of $T^*$ we must have 
$$ (T\alpha, \delta) = (\alpha, T^*\delta), \qquad \alpha,\delta \in V $$ 
So, in our case 
\begin{align*}
  (\alpha, T^*\delta) &= (T\alpha, \delta)\\
      &= \bigl((\alpha,\beta)\gamma, \delta\bigr)\\
      &= (\alpha, \beta)(\gamma,\delta)\\
      &= \bigl(\alpha, \overline{(\gamma,\delta)}\beta\bigr)\\
      &= \bigl(\alpha, (\delta, \gamma)\beta\bigr)
\end{align*}
so $T^*\delta = (\delta, \gamma)\beta$, all $\delta \in V$.