# Pick out true statements.

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)

With the inner product $<f,g>=\int^{\infty}_{-\infty} f(t)g(t) \,dt$ 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.
• @Math Man: Note that same idea works for $a$ and $b$ also. (+1) Jan 24, 2017 at 12:43