[Math] Orthogonal complement of a subspace in a Hilbert space

functional-analysishilbert-spaces

In this question it is stated that if $W$ is a closed subspace of $V$, we may define $W^\perp$ to be the subspace of all vectors $v \in V$ such that $\langle v | w\rangle =0$ for all $w \in W$.

Does the subspace really have to be closed to define the orthogonal complement? I believe it's possible to define the orthogonal complement of any subset.

Best Answer

You're absolutely right. You can just as easily define $S^\bot$ for any subset $S$ of your Hilbert space $V$.

$$S^\bot = \{ v \in V : \langle v \vert w \rangle = 0 \text{ for all } w \in S \}$$

You can also check that, no matter the structure of $S$,

  • $S^\bot$ is a closed subspace, and
  • $(S^\bot)^\bot$ is the smallest closed subspace containing $S$.
Related Question