inner product space questions

Question

I didn't succeed in solving these two short questions (in the link).

In the first one I think I need to work with bases, but I don't know how. In the second question I just don't know how to start.

Can you show me how to solve these?

Answer 1

First, prove that

$$(A+B)^\perp=A^\perp\cap B^\perp:$$

$$x\in (A+B)^\perp\implies \langle x,a+b\rangle=0\;,\;\;\forall\,a\in A\,,\,b\in B$$

Now, choose first $\;b=0\;$ and then $\;a=0\;$ in the above, getting

$$\begin{cases}\langle x,a\rangle=0\;,\;\;\forall a\in A\iff x\in A^\perp\\{}\\\langle x,b\rangle =0\;,\;\;\forall b\in B\iff x\in B^\perp\end{cases}\implies\;x\in A^\perp\cap B^\perp$$

I leave to you the second, opposite inclusion.

Finally, to obtain your first inclusion, put simply $\;W=A^\perp\;,\;\;U=B^\perp\;$ and pass to orthogonal complements.

Added on request:

$$\forall\,x=T^*v\in\text{Im}\,T\;,\;\;\forall\,u\in\ker T:$$

$$\langle u,x\rangle=\langle u,T^*v\rangle=\langle Tu,v\rangle=\langle0,v\rangle=0\implies u\in \left(\text{Im}\,T^*\right)^\perp$$