TITLE: How to show that a given set is an vector space?
QUESTION [5 upvotes]: I have the following set:
$$
\mathcal{V} = \left\{f: \int_{-\infty}^{\infty}f(x)^2 e^{-x^2}\,dx < \infty\right\}
$$
To show it is a vector space I need to show that:

addition and subtraction are defined
The set is closed under linear combinations
There is a zero vector.

However, I'm not sure how to show these properties with an integral.

REPLY [4 votes]: Generally, one just defines such operations pointwise:
$$(f\pm g)(x) = f(x)\pm g(x) \text{ and } (\alpha f)(x) = \alpha f(x).$$
Then,
$$\int (\alpha f \pm \beta g)^2(x)e^{-x^2} \ dx = \alpha^2\int f^2(x)e^{-x^2} \ dx + \beta^2\int g^2(x)e^{-x^2} \ dx \pm 2\alpha\beta\int f(x)g(x)e^{-x^2} \ dx.$$
You should be able to show that the third integral is finite via Hölder's inequality. The first two are finite by definition. For the zero vector take $f \equiv 0$.
Edit (to address the comment): The operations are defined pointwise and $(f+z)(0) \neq f(0)$, so $z$ is not the zero vector (ie, $f \equiv 0$ is the unique zero vector). The issue you raise wouldn't arise unless perhaps one wanted to put a norm on $\mathcal{V}$ like
$$\|f\|_\mathcal{V} := \int f^2(x)e^{-x^2} \ dx.$$
In that case, one could define an equivalence relation $\sim$ by $f \sim g$ if and only if $f=g\text{ a.e.}$ and then consider $\mathcal{V}/\sim$.