Let $\mathscr{P}_{l}$ denote the vector space of all polynomials of degree $\leq l$, with the norm $$||p||=\int_{-\infty}^{+\infty}|p(x)|e^{-|x|}dx.$$ I'm supposed to prove that this space is complete with regard to the corresponding metric. Is there a simple way to do this? Here's what I've got so far. Let $(p_{n})_{n \in \mathbb{N}}$ be a Cauchy sequence in $\mathscr{P}_{l}$, so then we have, for some arbitraty $\varepsilon>0$, and a sufficiently large $n_{0}$, for all $m, n \geq n_{0}$, $$\int_{-\infty}^{+\infty}|p_{n}(x)-p_{m}(x)|e^{-|x|}dx<\varepsilon$$ holds.
I want to prove that the difference between two coefficients of the same degree for $p_{n}, p_{m}$ becomes small, use that $\mathbb{R}$ is complete, and then get a polynomial $p_{\infty}$ which is determined by the values that the coefficients of the sequence converge to. I tried using the inequality $$\int_{-\infty}^{+\infty}|f(x)|dx \geq |\int_{-\infty}^{+\infty}f(x)dx|,$$ but this only works on even degree coefficients. Is this the right approach? Is there an easier way to do this?
