Showing that a vector space is complete

Question

Let be $ A$ the vector space of absolute converging fourier series with norm $ ||f||_A := \sum_{n \in Z } | \hat{f} (n) | $

I know that if $A$ is complete, than every cauchy sequence in $ A$ has a limit in $ A$. I am struggling with finding such cauchy sequence ..

thank you for any help !!

Answer 1

I suppose $A$ is supposed to be functions with absolutely convergent Fourier series. Let $(f_k)$ be a Cauchy sequence in $A$. Then $f_k(x)=\sum \hat {f_k} (n)e^{inx}$ uniformly for each $k$. Hence $\sup_x |f_k(x)-f_j(x)| \leq \sum | \hat {f_k} (n)-\hat {f_j} (n)| \to 0$ as $j,k \to \infty$. Thus $f_k$ converges uniformly to a function $f$ (which is, of course, continuous and periodic). By uniform convergence we get $\hat {f_k} (n) \to \hat {f} (n)$ for all $n$. Also $\sum |\hat {f} (n)| <\infty$ by Fatou's Lemma. Can you now see that $\|f_k-f\|\to 0$?