The Orthogonal Complement

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.

 

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

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

 

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

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

 

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

 

How did you guess that it is the sum of?

since my goal is to reach this equality.

How can we prove it?

 

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

 

Oh yes, thats the problem. But the problem now how can we prove that
[itex](M\cap N)^\bot=M^\bot+N^\bot[/itex]

 

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