0

are there any theorems that give us any conditions to know if a linear subspace $E$ plus its orthogonal complement span the whole vector space?

For exemple, I know that in $\mathbb{R}[X]$, the complement orthogonal of the hyperspace $Span(1+X, 1+X^2, ...)$ is $\{0\}$ and thus the sum does not span the whole space

Thanks !

poloC
  • 191
  • 2
    In order to define "orthogonal complement" you need an inner product, or at least a bilinear form. Which bilinear form are you using on $\mathbb R[X]$? – Robert Israel Sep 06 '18 at 20:38

1 Answers1

0

Hint: you can prove that $(A + B)^T=A^T \cap B^T$, this is rather easy. Then it is sufficient to notice that $(A^T+A)^T=A^T \cap A=0$.

Michał Zapała
  • 1,630