Question

Let \(S\) be a subset of an inner product space \(X\). Show that \(S^{\perp}\) is a closed linear subspace of \(X\).

Answer 1

I just need the closed part @eliassaab

Answer 2

Let z be in the closure of \( S^{\perp} \), the ther is a sequence \( z_n\in S^{\perp}\) converging to z, let \( x\in S\) then
\[
<x, z> = < x, \lim_{n\to \infty} z_n>=\lim_{n\to \infty} < x, \ z_n>=0

\]

Answer 3

man, you are good at this:)

Answer 4

everytime you answer something you make it seem trival

Answer 5

Meh, Math Majors.

Answer 6

@zzr0ck3r Thanks