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?