How to show that a vector space is closed?

Question

I am trying to complete a proof which requires me to prove that a subspace $H$ of $L^2{(\Omega,\mathcal{F}},\mathbb{P})$ is closed vector space in $L^2{(\Omega,\mathcal{F}},\mathbb{P})$

What do I need to show in order to prove this?

Answer 1

It sounds like you are focused on the closedness of $H$ rather than the subspace part, so I'll address just the closedness. (Note that if you've already established $H$ is a subspace, then $H$ is necessarily a vector space itself.)

Let $\{x_n\}$ be a Cauchy sequence in $H$ (Cauchy with respect to whatever norm you have on $H$). Then $x_n\to x$ for some $x$. Show that $x\in H$.