$V$ is a closed subspace of $L^2[0,1]$; $f(x)=x$ & $g(x)=x^2$ are in $L^2[0,1]$. Orthogonal complement of $V = span(f)$. Find the the orthogonal projection $Pg$ of $g$ on $V$. (Verification: if $(g-Pg)(x)=3x/4 $ for $x$ in $[0,1]$ then the $Pg$ is correct.)
Orthogonal complement of $V = span(x)$. Find the the orthogonal projection of $x^2$ on $V$.
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linear-algebra
functional-analysis
1 Answers
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We have
$L^2[0,1]=V \oplus V^{\perp}$ and $P$ is defined as follows: if $\phi \in L^2[0,1]$ then there is $a \in \mathbb R$ and $h \in V^{\perp}$ such that $\phi=af+h$. Then:
$$P \phi:=h.$$
Now, if $\phi =g$, then find $h \in L^2[0,1]$ and $a \in \mathbb R$ such that
$x^2=ax+h(x)$ for all $x \in [0,1]$ and $\int_{0}^1xh(x) dx=0$