Prove if $X$ is in closed linear subspace of the Hilbert space $L^{2}(\Omega, \mathcal{F}, P)$, then $X_{+}$ is also in it.

Question

I am trying to prove the following, if $X\in Y$, where $Y$ is a closed linear subspace of the real Hilbert space $L^{2}(\Omega, \mathcal{F}, P)$, then $X_{+}$ is also in Y. ($X_{+}$ is just the positive part of $X$) Thanks in advance.

Answer 1

You can't prove it, since its not true. Suppose that $f\in L^2(\Omega,\mathcal F,P)$ is such that $f\neq f_+$ and that $f\neq f_-$. Take $Y=\mathbb Rf$. Then $Y$ is closed, $f\in Y$, and $f_+\notin Y$.