Pick out true statements.

Question

Which of the following statements are true?

a. Let $g\in C[0,1]$ be fixed. Then the set $$A=\{f \in C[0,1] | \int^1_0 f(t)g(t) \,dt=0\}$$ is closed in $C[0,1]$.

b. Let $g\in C_c(\mathbb{R})$ be fixed. Then the set $$A=\{f \in C_c(\mathbb{R}) | \int^{\infty}_{-\infty} f(t)g(t) \,dt=0\}$$ is closed in $C_c(\mathbb{R})$.

c. Let $g\in L^2(\mathbb{R})$ be fixed. Then the set $$A=\{f \in L^2(\mathbb{R}) | \int^{\infty}_{-\infty} f(t)g(t) \,dt=0\}$$ is closed in $L^2(\mathbb{R})$.

For option a) and b) pick a sequence $\{f_n\}$ in respective $A$ and we can show that $\lim f_n \in A$ by interchanging integral and the limit sign by DCT. I'm not sure about option c)

Answer 1

With the inner product $=\int^{\infty}_{-\infty} f(t)g(t) \,dt$ the space $H=L^2( \mathbb R)$ is a Hilbert space.

We put $U= \{tg: t \in \mathbb R\}$. Then $U$ is a $1$ -dimensional subspace of $H$, hence closed, and we have $A=U^{\perp}$. Therefore $A$ is closed.