The problem statement, all variables and given/known data.
Let the ##\mathbb R##- vector space ##C([-1,1])=\{f:[-1,1] \to \mathbb R\ : f \space \text{is continuous}\ \}## with the inner product ##<f,g>=\int_{-1}^1 f(t)g(t)dt##. Determine the orthogonal complement of the subspace of even functions (call that subspace ##S##).
The attempt at a solution.
Straight from the definition of orthogonal complement, ##g \in (C([-1,1]))^\bot## if and only if ##\int_{-1}^1 g(t)f(t)dt=0## for all ##f## even function. By hypothesis, ##f(-t)=f(t)##, so ##\int_{-1}^1 g(t)f(-t)dt=0## From this condition I don't know what else to conclude. I would appreciate any suggestions.
Let the ##\mathbb R##- vector space ##C([-1,1])=\{f:[-1,1] \to \mathbb R\ : f \space \text{is continuous}\ \}## with the inner product ##<f,g>=\int_{-1}^1 f(t)g(t)dt##. Determine the orthogonal complement of the subspace of even functions (call that subspace ##S##).
The attempt at a solution.
Straight from the definition of orthogonal complement, ##g \in (C([-1,1]))^\bot## if and only if ##\int_{-1}^1 g(t)f(t)dt=0## for all ##f## even function. By hypothesis, ##f(-t)=f(t)##, so ##\int_{-1}^1 g(t)f(-t)dt=0## From this condition I don't know what else to conclude. I would appreciate any suggestions.
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