I want to prove the norm $\|g\| = max_{x\in R }|g(x)|$ is complete in the space ${g(x)=\int_{-c}^c F(w)e^{iwt}dw: F(w)\in L^2[-c,c]}$. But I do not know how to prove that the Cauchy sequence $g_n$ converge to the space.
Because of completeness of numbers, there is a $g(t)$ such that $g_n(t)\to g(t), n\to \infty, t\in R$. Consider $g(t), t\in R$. Cauchy Sequence: $\forall a>0$, there is a number N, such that $\|g_n-g_m\|<a, \forall n,m>N$. That is, $|g_n(t)-g_m(t)|<a,\forall n,m>N, \forall t\in R$ Let $m\to\infty$, we have $g_n\to g$. Then I do not know how to prove that the $g\in \{g(x)=\int_{-c}^c F(w)e^{iwt}dw: F(w)\in L^2[-c,c]\}$. Please help me.