Linear Algebra – Prove Projection is Self Adjoint if Kernel and Image are Orthogonal Complements

Question

Let $V$ be an IPS and suppose $\pi : V \to V$ is a projection so that $V = U \oplus W$ (ie $ V = U + W$ and $U \cap W = \left\{0\right\}$) $ \ $ where $U = \ker(\pi)$ and $W = \operatorname{im}(\pi)$, and if $v = u + w \ $ (with $u \in U, \ w \in W$) then $\pi(v) = w$.
Prove $\pi$ is self adjoint if and only if $U$ and $W$ are orthogonal complements.

I'm hoping someone can give me a few hints on how to begin this question.

Answer 1

$\pi$ self-adjoint

$\iff \forall x, y \in V, \langle \pi(x)\mid y\rangle=\langle x\mid \pi(y)\rangle$

$\iff \forall x_U,y_U\in U, \forall x_W,y_W \in W, \langle \pi(x_U+x_W)\mid y_U+y_W\rangle=\langle x_U+x_W\mid \pi(y_U+y_W)\rangle$

$\iff \forall x_U,y_U\in U, \forall x_W,y_W \in W, \langle x_W\mid y_U+y_W\rangle=\langle x_U+x_W\mid y_W\rangle$

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$\iff \forall x_U,y_U\in U, \forall x_W,y_W \in W, \langle x_W\mid y_U\rangle+\langle x_W\mid y_W\rangle=\langle x_U\mid y_W\rangle+\langle x_W\mid y_W\rangle$

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$\iff \forall x_U,y_U\in U, \forall x_W,y_W \in W, \langle x_W\mid y_U\rangle=\langle x_U\mid y_W\rangle$

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$\iff \forall y_U\in U, \forall x_W \in W, \langle x_W\mid y_U\rangle=0$