I was trying to find an orthogonal complement to a subset of $L_{2}(0,1)$. And I come up with this question.
If I know that $u,f\in L_{2}(0,1)$, $\int_{0}^{1} u(x)dx=0$, and $\int_{0}^{1}f(x)\overline {u(x)}dx=0$, can I say something about $f(x)$, i.e. can I formulate a general set about $f(x)$ satisfying these conditions?
Any help is appreciated!
Find a set of functions satisfying integration to 0
1
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integration
functional-analysis
orthogonality
1 Answers
1
All you know about $f$ is contained in the statements $f \in L_2(0,1)$ and $\int_0^1 f(x) \overline{u(x)}\; dx$, i.e. $f$ is in the orthogonal complement of the function $u$. This is a closed subspace of codimension $1$. Namely, for any $g \in L^2(0,1)$, $g - c u$ is in this subspace, where $c = \int_0^1 g(x) \overline{u(x)}\; dx/\int_0^1 |u(x)|^2\; dx$.