1. ## Orthogonal Complements

$\displaystyle U \subseteq R^n$ and $\displaystyle U^\perp$ is it's orthogonal complement.

Is there a case where

$\displaystyle {(U^\perp)}^\perp \neq U$

Thanks.

2. Originally Posted by jayshizwiz
$\displaystyle U \subseteq R^n$ and $\displaystyle U^\perp$ is it's orthogonal complement.

Is there a case where

$\displaystyle {(U^\perp)}^\perp \neq U$

Thanks.

Not in finite dimensional cases.

Tonio

3. Thanks.

But is there a case where

$\displaystyle ((U^{\perp}){^\perp}){^\perp} \neq U{^\perp}{$

It would seem to me that I would reach the same conclusion as before...

4. U need not be a subspace, so choose a subset. The orthogonal complement must be a subspace, so the orthogonal complement of the orthogonal complement would be the subspace spanned by U, which will not be U if U is not a subspace.

5. I don't exactly understand what you wrote. Is there a difference between the first case i presented and this new one? Thanks.

6. Yes.
In your first case, you chose $\displaystyle U \subseteq R^n$ such that U may or may not be a subspace. By definition of orthogonal complement, $\displaystyle U^\perp$ is a subspace. So $\displaystyle {(U^\perp)}^\perp$ is also a subspace (of $\displaystyle R^n$), and we know $\displaystyle U \subseteq U^\perp$ and in fact $\displaystyle U^\perp$ is the subspace spanned by U.
For your second case, you chose to start with $\displaystyle U^\perp$ which is a subspace of $\displaystyle R^n$, so $\displaystyle {({(U^\perp)}^\perp)}^\perp$ is the subspace spanned by $\displaystyle U^\perp$, but that implies equality based on the definition of spanned.