Completeness of a subspace of $(C[0,1],L^1)$

Question

I'm trying to prove the following statement

Consider the metric space $C[0,1]$ with $L^1$ norm. Let $A:=\{f\in C[0,1] : f(x)=a\sin(x)+b\cos(x) , a,b\in\mathbb{R}\}$. Is $A$ complete?

I think it's complete but I cannot prove it. I though it would be similar to the case of $B=\{g\in C[0,1] : g(x)=a\sin(x), a\in\mathbb{R}\}$, but a Cauchy sequence on $A$ doesn't give me (easily) a Cauchy sequence on the coefficients (That was the easy part of $B$). Any suggestions?

Answer 1

You can easily show that $$A= \{ f\in C([0, 1]) : f(x)=a\sin(x+b), \, a\in\mathbb{R}, \, b\in\mathbb{R} \} $$ Could you show that $A$ is complete now?

Answer 2

Hint:

Notice that $A$ is a finite-dimensional subspace of the vector space $C[0,1]$. Namely, $$A = \operatorname{span}\{\sin, \cos\}$$

Any finite-dimensional normed space is complete.