Is there an analog of De Morgan's laws for linear spans of vector spaces?

Question

If I am given two linear subspaces $W_1$ and $W_2$ of a vector space $V$, is there a theorem such as De Morgan's law, stating something like \begin{equation} W_1+W_2=(W_1^\perp\cap W_2^\perp)^\perp. \end{equation} I'm not sure if this is exactly the correct way to write such a statement, but it seems reasonable, and I don't feel like reinventing the wheel. I would just like to know what the theorem is called (if it is true) or a good reference if it doesn't have a name. I would also like the generalization to more than two subspaces if possible.

Answer 1

We can show that for subspaces $W_1,W_2\subseteq V$, we have: $$(W_1+W_2)^{\perp} = W_1^{\perp}\cap W_2^{\perp}.$$

Let $v\in W_1^{\perp}\cap W_2^{\perp}$, and let $w_1+w_2\in W_1+W_2$ where $w_i\in W_i$ for $i=1,2$. Then, we have $\langle v,w_1\rangle=0$ and $\langle v,w_2\rangle=0$, and as $\langle-,-\rangle$ is bilinear, it follows that $$0 = \langle v,w_1\rangle+\langle v,w_2\rangle = \langle v,w_1+w_2\rangle.$$ Hence, $v\in (W_1+ W_2)^{\perp}$.

Now, let $u\in(W_1+W_2)^{\perp}$, and again let $w_i\in W_i$ for $i=1,2$. Then in particular, $w_1\in W_1+W_2$ and so $\langle u,w_1\rangle=0$. By the exact same reasoning, $\langle u,w_2\rangle=0$. Hence, $u\in W_1^{\perp}$ and $u\in W_2^{\perp}$, and therefore $u\in W_1^{\perp}\cap W_2^{\perp}$.

Lastly, if we have a vector space $V$ with subspaces $W_1,...,W_n$, an application of induction gives $$(W_1+\cdots+W_n)^{\perp} = W_1^{\perp}\cap\cdots\cap W_n^{\perp}.$$