Orthogonal complement of orthogonal complement

Question

Let $U$ be a subspace of $V$ (where $V$ is a vector space over $C$ or $R$). The orthogonal complement of the orthogonal complement of $U$ is not equal to $U$ in general (equal only for dim $V$ finite).

Can anyone give me a simple example when the orthogonal complement of the orthogonal complement of $U$ is not $U$.

Any non closed subspace of a Hilbert space. – egreg Jun 12 '13 at 10:38 — egreg, Jun 12 '13 at 10:38

DonAntonio · Accepted Answer · 2013-06-12T10:37:02.873

6

Let

$$V:=\left\{ f:[0,1]\to\Bbb R\;;\; f\;\;\text{is continuous}\right\}\;\;\text{over}\;\;\Bbb R$$

and with the inner product

$$\langle f,g\rangle:=\int\limits_0^1f(x)g(x)dx$$

Let

$$U:=\{ f \in V\;;\;f(0)=0\}\implies U^\perp=\{0\}\;,\;\;U^{\perp\perp}=\ldots$$

edited Jun 12 '13 at 10:37

answered Jun 12 '13 at 10:30

DonAntonio

214,715

Is this an inner product? The function $f(0)=1$, $f(x)=0$ for $0<x\le1$ is Riemann integrable and non zero, while $\langle f,g\rangle=0$ for all $g$. – egreg Jun 12 '13 at 10:36
Sorry, wrong word from another problem I was addressing. Thanks, edited. – DonAntonio Jun 12 '13 at 10:37
Is it trivial to see that $U^\perp={0}$ ? – Dark Jun 22 '16 at 14:51

Orthogonal complement of orthogonal complement

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