Let $ \ f,g: \mathbb{R} \to \mathbb{R} \ $ be continuous such that $ \ \max \{\int_{-\infty}^{\infty} |f| dx, \ \int_{-\infty}^{\infty} |g| dx \} \ < \infty $.
(i) If $ \ \lim_{|x| \to \infty} f(x)=0=\lim_{|x| \to \infty} g(x) \ $ , then $ \ |f|^2 , \ |g|^2 \ $ also have finite integrals.
(ii) Prove that linear combination of $ \ f,g \ $ forma an Euclidean space with the usual scalar product.
Answer:
(i)
Let $ \ f(x)=e^{-|x|}=g(x) \ $.
Then , we have
$ \ \lim_{|x| \to \infty} f(x)=0=\lim_{|x| \to \infty} g(x) \ $
Further,
$ \int_{-\infty}^{\infty} |f|^2dx \leq \int_{-\infty}^{\infty} e^{-2|x|} dx \ =2 \int_{0}^{\infty}e^{-2|x|} dx=1 $
Thus $ \ |f|^2 \ $ has finite integral.
Similarly,
$ |g|^2 \ $ also has finite integral.
But I can not prove it for general functions.
Help me with the general proof.
Since $f(x) \to 0$ as $\lvert x \rvert \to \infty$, there is an $M$ so that $\lvert f(x) \rvert <\varepsilon$ for $\lvert x \rvert > M$. Also, since $\lvert f \rvert$ is continuous on $[-M,M]$, it is bounded, by $K$, say. But then $$ \int_{-\infty}^{\infty} \lvert f \rvert^2 \, dx = \int_{|x| \leq M} \lvert f \rvert^2 \, dx + \int_{|x| > M} \lvert f \rvert^2 \, dx \leq K\int_{|x| \leq M} \lvert f \rvert \, dx + \varepsilon \int_{|x| > M} \lvert f \rvert \, dx \leq (K+\varepsilon) \left( \int_{|x| \leq M} \lvert f \rvert \, dx + \int_{|x| > M} \lvert f \rvert \, dx \right) = (K+\varepsilon)\int_{-\infty}^{\infty} \lvert f \rvert \, dx < \infty $$ by the assumption.