let V be a closed subspace of L^2[0,1](i.e.,space of square Integrable functions defined on [0,1]) and let f,g ∈L^2[0,1] such that f(x)=x and g(x)=x^2. Suppose orthogonal complement of V =span {f} and P_g is the orthogonal projection of g on V,then (g - P_g)(x) is
(A)3/4*x (B) 1/4*x (C) 3/4*x^2 (D) 1/4*x^2
To calculate the projection onto span($f$), you should actually be using $\langle \tilde f, g \rangle \tilde f$, where $\tilde f = f / \|f\|_2$ is normalized. Here, $\|f\|_2^2 = 1/3$.
Also, note in the problem statement the orthogonal complement of $V$ is given by span($f$). So, $P_g$ is actually given by a different formula, but maybe $g - P_g$ is easier to calculate...
