• Support PF! Buy your school textbooks, materials and every day products Here!

Caracterizing a subspace of L^2

  • Thread starter quasar987
  • Start date
quasar987
Science Advisor
Homework Helper
Gold Member
4,771
7
[SOLVED] Caracterizing a subspace of L^2

1. Homework Statement

Call M the subspace of L²([0,1]) consisting of all functions of vanishing mean. I.e., [itex]u\in M \Rightarrow \int_0^1u(s)ds=0[/itex].

I am trying to find the dimension of the orthogonal of M,

[tex]M^{\perp}=\{x\in L^2([0,1]):\int_0^1x(s)u(s)ds=0 \ \forall u\in M\}[/tex]

I would be surprised if [itex]M^{\perp}[/itex] was anything other than the constant functions, but my attempts at a proof have been unsuccessful.

Any idea how to go at this?
 

Answers and Replies

quasar987
Science Advisor
Homework Helper
Gold Member
4,771
7
Solved.
 
Last edited:
morphism
Science Advisor
Homework Helper
2,013
4
Was it the (a.e.) constant functions in the end?

Sorry I couldn't help -- been busy with studying for my finals!
 
Dick
Science Advisor
Homework Helper
26,258
618
Was it the (a.e.) constant functions in the end?

Sorry I couldn't help -- been busy with studying for my finals!
How could it be anything else? The set of vanishing mean functions is basically defined as the set of all u(x) such that <u(x),1>=0. L^2 is a norm up to a.e.
 
morphism
Science Advisor
Homework Helper
2,013
4
Well, yeah... :tongue2:
 
Dick
Science Advisor
Homework Helper
26,258
618
Well, yeah... :tongue2:
Nice to talk to an erudite and sophisticated gentleman for a change. Instead of a bonehead. :)
 

Related Threads for: Caracterizing a subspace of L^2

  • Last Post
Replies
1
Views
699
Replies
8
Views
2K
Replies
4
Views
4K
Replies
2
Views
3K
Replies
3
Views
833
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
3
Views
760
Replies
24
Views
2K
Top