I have, one more time, a question about how to determine an orthographic projection on a closed subspace of $H$, where $H$ is a Hilbert space. Actually, we're considering the closed subspace : $E= \{ f \in L^1([-1;1]) \; | \; \int_{-1}^{1} xf(x) \mathrm{d}x = 0 \} $, and we want to :
1) give the orthographic projection of $f(x) = x+x^2cos(x)$ on $E$
2) Show that : $F=\{ f \in L^2([-1;1]) \; | \; \int_{-1}^{1}|x^{-1}f(x)| \mathrm{d}x < \infty $ and $ \int_{-1}^{1} x^{-1}f(x) \mathrm{d}x = 0 \} $ is a dense subset.
For the first question, this is what I do : we can note that $x \rightarrow x \in E^{\perp}$, so as $f(x) = x+x^2cos(x)$ with $x^2cos(x) \in E$, we have that the orthographic projection of $f$ on $E$ is $x^2cos(x)$. Am I right ?
But for the second question, I don't know how to do... Probably I have to show that $F^{\perp} = \{0\}$, but I don't figure out how. I've tried to take an element $g$ on $F^{\perp}$ and show $g(x) = 0$ almost everywhere, but I don't succeed to do that.
So, if someone could help me, thank you !
Your answer to 1) is right. For 2) take any $f \in L^{2}[-1,1]$ and first approximate it by a function that vanishes in some open interval around $0$. ($fI_{\{x:|x| >1/n\}} \to f$ in $L^{2}[-1,1]$). So we may as well assume that $f$ vanishes near $0$. Let $g_n(x)=f(x)-cI_{(\frac 1 {2n},\frac 1 n)}$ where $c$ is such that $\int_{-1}^{1} \frac {f(x)} x \, dx=c \int_{-1}^{1} \frac 1 x I_{(\frac 1 {2n},\frac 1 n)}dx$ or $c \log \, 2=\int_{-1}^{1} \frac {f(x)} x\, dx$. Then $g_n \in E$ for all $n$. Note that $\|g_n-f\|_2=|c| \|I_{(\frac 1 {2n},\frac 1 n)}\|_2 \to 0$. Hence $E$ is dense in $L^{2}[-1,1]$.