Prove that Lei-Lin space is a Banach space

Question

The following space is called Lei-Lin space: $$\mathcal{X}^{-1}=\{f\in \mathcal{D}'(\Bbb{R}^n),~\hat{f}\in L_{loc}^1(\Bbb{R}^n)~\int_{\Bbb{R}^n}|\xi|^{-1}|\hat{f}(\xi)|d\xi<\infty~\},$$ where $\hat{f}$ stands for the Fourier transform of $f$, the space is endowed with its natural norm $$\|f\|_{\mathcal{X}^{-1}}=\int_{\Bbb{R}^n}|\xi|^{-1}|\hat{f}(\xi)|d\xi$$ I want to prove that $\mathcal{X}^{-1}$ is a Banach space, in other words: prove that any Cauchy sequence in $\mathcal{X}^{-1}$ is a convergent one in $\mathcal{X}^{-1}$: i already know that saying $f_k$ is a Cauchy sequence in $\mathcal{X}^{-1}$ means that $\hat{f}_k$ is Cauchy sequence in $L^1(|\xi|^{-1},\Bbb{R}^n)$ and as $L^1$ is complete then there exists $l(\xi)$ that belongs to $L^1(|\xi|^{-1},\Bbb{R}^n)$ such that $\hat{f}_k$ $\to$ $l$ in $L^1(|\xi|^{-1},\Bbb{R}^n)$; but how can I prove that $l(\xi) $ is in fact the Fourier transform of a function $f(x)\in \mathcal{X}^{-1}$ such that $f_k\to f$ in $\mathcal{X}^{-1}$ as $k \to \infty$?