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))

1 Answer 1

  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 $U$.
  • $\begingroup$ 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

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