# Caracterizing a subspace of L^2

1. Apr 5, 2008

### quasar987

[SOLVED] Caracterizing a subspace of L^2

1. The problem statement, all variables and given/known data

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

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

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

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

Any idea how to go at this?

2. Apr 8, 2008

### quasar987

Solved.

Last edited: Apr 8, 2008
3. Apr 8, 2008

### morphism

Was it the (a.e.) constant functions in the end?

Sorry I couldn't help -- been busy with studying for my finals!

4. Apr 8, 2008

### Dick

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.

5. Apr 8, 2008

### morphism

Well, yeah... :tongue2:

6. Apr 8, 2008

### Dick

Nice to talk to an erudite and sophisticated gentleman for a change. Instead of a bonehead. :)