0
$\begingroup$

Let V be a real vector space with a topology induced from an inner product. Let S be a subset. Can we say either of the following ?

  1. closure(S) = perp(perp(S)) , [where perp(X) = orthogonal complement of X]
  2. closure(span(S)) = perp(perp(S))
$\endgroup$

1 Answer 1

1
$\begingroup$
  1. is false: the orthogonal complement $S^\perp=\{x\mid\forall s\in S: x\perp s\}$ of any set $S$ is always a closed subspace, whereas the closure need not be a subspace.
  2. is true: prove $S^\perp =\overline{\mathrm{span}(S)}^\perp$, and apply $U^{\perp\perp} =U$ for a closed subspace $U$.
$\endgroup$
2
  • $\begingroup$ what do you mean the closure need not be? The closure is by definition closed isn't it? $\endgroup$ Dec 7, 2019 at 14:48
  • 1
    $\begingroup$ Need not be a subspace. For example, a closed ball is closed but not a subspace. Its perp is $\{0\}$. $\endgroup$
    – Berci
    Dec 7, 2019 at 15:00

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.