Let V be a finite dimensional real inner product space and U, W subspaces of V such that U is orthogonal to W. Show that for any $v ∈ V$ $$||v||^2 ≥ ||proj U (v)||^2 + ||projW (v)||^2$$
Hello guys. I am trying to prove this. I am doing 3 cases. First one is v is in span of orthogonal basis of U. Second one is that v is in span of orthogonal basis of W. These two case is clear. But in the final case, when v is not element of span of these two orthogonal bases, i could not find a way to prove it.
You can combine your two cases into one by producing the orthogonal complement of $U \oplus W$ inside $V$.
Indeed, if $V$ is the orthogonal direct sum $U_1 \oplus \cdots \oplus U_n$, then it is generally true for $ v = u_1 + \cdots + u_n$, that
$$\|v\|^2 = \|u_1\|^2 + \cdots \|u_n\|^2.$$
Apply this principle to your case, with $V = U \oplus W \oplus [U \oplus W]^{\perp}$.