Orthonormal Basis of Two Subspaces

Question

Let $S,T$ $\subset \mathbb{R^n}$

Prove that it is possible to choose an Orthonormal Basis W for S and W' for T such that

$W = (s_1,....,s_k)$

$W' = (t_1,.....t_m)$

$<s_i,t_j>$ = 0 if $i \neq j$

$<s_i,t_j>$ $ \; \geq$ 0 if $i = j$

Note : $<.>$ defines the Inner Product on $\mathbb{R^n}$

Is it possible to use Gram Schmidt here?

Answer 1

Have you heard about biorthonormal system?