What Are the Orthogonal Complements of Union and Intersection in Hilbert Spaces?

[LikeMath]

Let M and N be two subsets of a hilbert space H.
What are orthogonal complements of following sets:
1) The union of M and N.
2) The intersection of M and N.

 

[micromass]

So, what did you try already??

 

[LikeMath]

x\in (M\cap N)^\bot means
<x,y>=0 for all y\in M \text{ and } y\in N and hence

x\in M^\bot\cap N^\bot.. The other direction is obvious, so we get (M\cap N)^\bot=M^\bot\cap N^\bot
This is for the intersection, but I strongly think that I had a mistake.

 

[LikeMath]

x\in (M\cap N)^\bot means
<x,y>=0 for all y\in M \text{ and } y\in N and hence

x\in M^\bot\cap N^\bot.. The other direction is obvious, so we get (M\cap N)^\bot=M^\bot\cap N^\bot
This is for the intersection, but I strongly think that I had a mistake.

 

[micromass]

I don't think the other inclusion is obvious.

In fact, I suspect (M\cap N)^\bot = M^\bot + N^\bot...

 

[LikeMath]

I do not know what do you mean by the sum of tow sets? and how can I prove that?

 

[Deveno]

one can't, in general, sum two sets, but if we already have an addition defined, then:

A + B = {a+b : a is in A, b is in B}.

 

[LikeMath]

How did you guess that it is the sum of?

since my goal is to reach this equality.

How can we prove it?

 

[Deveno]

normally, to prove 2 sets are equal, you show they contain each other.

 

[LikeMath]

Of course, but I did not manege tp solve it!
On the other hand, (M\cap N)^\bot=M^\bot+N^\bot means M^\bot+ N^\bot\subset M^\bot\cap N^\bot, but this seems senseless, does not it?

 

[Deveno]

x\in (M\cap N)^\bot means
<x,y>=0 for all y\in M \text{ and } y\in N and hence

x\in M^\bot\cap N^\bot.. The other direction is obvious, so we get (M\cap N)^\bot=M^\bot\cap N^\bot
This is for the intersection, but I strongly think that I had a mistake.

your definition of (M\cap N)^\bot isn't correct here.

we don't know that <x,y> = 0 for all y in M and y in N, just those y that are in the intersection. if y is in M-N, all bets are off.

 

[LikeMath]

Oh yes, that's the problem. But the problem now how can we prove that
(M\cap N)^\bot=M^\bot+N^\bot

 

[LikeMath]

(M\cap N)^\bot=M^\bot+N^\bot

I think we should have some contraintes on M and N, do not I?

 

[micromass]

I think we should have some contraintes on M and N, do not I?

Yeah, M and N should both be subspaces.