Question:
Suppose $C[−1, 1]$ is the vector space of continuous real-valued functions on the interval $[−1, 1]$
with inner product given by $\langle f, g\rangle = \int_{a}^b f(x)g(x)dx$
Let $U = {f ∈ C[−1, 1] : f(0) = 0}$ be the subspace of $C[−1, 1]$. Which of the following
statement(s) is(are) correct?Justify your answer.
(a) $C[−1, 1] = U ⊕ U^\bot$
(b) $U^\bot = \{0\}$
(c) $U^\bot$ is a proper and non-trivial subspace of $C[−1, 1]$
Difficulty: I am sure that option (b) is correct but not able to write a proof of it. Also about option (a) is valid for finite-dimensional subspace but it is not am I correct about it.
Best Answer
Hint: If $g \in U^{\perp}$ then $\int fg=0$ for all $f \in U$. Let $f_n(x)=g(x)$ for $|x| >\frac1 n$, $f_n(0)=0$ and $f_n$ have a straight line graph in $[-\frac 1 n ,0]$ as well as $[0, \frac 1 n]$. Then $\int g f_n=0$ and letting $n \to \infty$ yields $\int g^{2}=0$. Hence $g=0$. This proved b).
It is obvious from (b) that (a) is false. The constant function $1$ is in LHS but not in RHS. (c) is also answered by (b).