Prove that transformation is unitary transformation

Question

I have searched here but I couldn't find the answer: Let $T:V\to V$ be a unitary transformation in inner product space $V$, and let $W \subseteq V$ be subspace such that $T(w) \subseteq W$. I need to prove that the restriction $T_w:W\to W$ is also unitary transformation. Now from what I have searched I know that $T_w$ is diagonalizable so $T_w$ has othonormal basis consist of eigenvectors of $W$. How can I proceed from here to show that $T_w$ is unitary??? Thanks!!!

Answer 1

If we have inner product space $(V,\left<\cdot,\cdot\right>)$, then $T\colon V\to V$ is unitary iff $\left<Tx,Ty\right>=\left<x,y\right>$ forall $x,y\in V$

So $T_{|W}\colon W\to W$ is unitary on the inner product space $(W,\left<\cdot,\cdot\right>_{|W})$ because $$\left<T_{|W}x,T_{|W}y\right>_{|W}=\left<Tx,Ty\right>=\left<x,y\right>=\left<x,y\right>_{|W}$$ forall $x,y\in W$.