TITLE: Passing limit inside integral for functions in $L^1+L^2$ norm
QUESTION [5 upvotes]: Let $f\in L^1(\mathbb{R})\cap L^2(\mathbb{R})$, and let $f_k$ be functions in the Schwartz class such that $\|f-f_k\|_1+\|f-f_k\|_2\rightarrow 0$ as $k\rightarrow\infty$. Define $$g_k(t)=\int_\mathbb{R}f_k(x)e^{-itx}dx \text{    and    } g(t)=\int_\mathbb{R}f(x)e^{-itx}dx$$ Show that $\lim_{k\rightarrow\infty}g_k(t)=g(t)$ for all $t$.

I want to use something like dominated convergence thm to pass the limit inside the integral. But here it is not clear, because we only have convergence in $L^1+L^2$ norm. We don't know whether $f_k(x)\rightarrow f(x)$ at point $x$. What can we do?

REPLY [2 votes]: For every fixed $t$, the functional $\Phi_t(f) =   \int_{\mathbb R} f(x)e^{-itx}\,dx$ is  continuous with respect to  the $L^1$ norm:
$$|\Phi_t(f)-\Phi_t(h)|\le \int_{\mathbb R}|f(x)-h(x)|\,dx = \|f-h\|_{L^1}$$
Hence, the convergence $f_k\to f$ in $L^1$ norm implies $\Phi_t(f_k)\to \Phi_t(f)$. The additional condition about $L^2$ norm is not needed.