Orthogonality of the sum of two subspaces

Question

Let C and D be subspaces of vector space V. Now I understand C $\subseteq C+D$, however I don't see how come (C+D)$^\perp$ $\subseteq C$$^\perp$ and (C+D)$^\perp$ $\subseteq D$$^\perp$, where (C+D)$^\perp$ is the orthogonal complement of the set (C+D). Would anyone care to explain? Thank you.

Answer 1

Highlights:

$$x\in(C+D)^\perp\implies \langle x,y\rangle =0 \;\;\forall\,y\in C+D$$

And since $\;C\subset C+D\;$ , the above is true in particular for $\;c\in C\le C+D\;$:

$$\langle x,c\rangle=0\implies x\in C^\perp$$