Adjoint of projection onto direct sum of Hilbert spaces Let $K_n$ be Hilbert spaces and define \begin{equation*} K := \bigoplus_{\ell_2} K_n = \left\{ (x_1,x_2,\ldots) \in \bigoplus_{n=1}^\infty K_n : \sum_{n=1}^\infty \|x_n\|^2 < \infty \right\} \end{equation*}
It is easy to see that $K$ is a Hilbert space with the inner product $(x,y) = \displaystyle{\sum_{n=1}^\infty (x_n,y_n) } $. I'm interested in the adjoint of the operator projection \begin{equation*} \begin{split} & \pi_n :\bigoplus_{\ell_2}K_n \rightarrow K_n \\ & (x_1, \ldots,x_n, \ldots) \mapsto x_n. \end{split} \end{equation*} My idea is that ${\pi_n}^*$ is the natural inclusion from $K_n$ to $K$, but I can't find the correct way to prove it. Is my idea right? How can I prove it?
Than you for your help.

You are right, the adjoint of $\pi_n : K \to K_n$ is the canonical inclusion $\iota_n : K_n \to K$.
Let $(x_1, x_2, \ldots) \in K$ and $y_n \in K_n$. We have:
\begin{align}
\Big\langle \pi_n(x_1, x_2, \ldots), y_n\Big\rangle_K &= \langle x_n, y_n\rangle_{K_n} \\
&= \Big\langle (x_1, x_2, \ldots), (\underbrace{0, \ldots, 0}_{n-1}, y_n, 0, \ldots)\Big\rangle_{K}\\
&= \Big\langle (x_1, x_2, \ldots), \iota_n(y_n)\Big\rangle_{K}\\
\end{align}
Therefore $\pi_n^* = \iota_n$.
