# “vector spaces” without additive inverses

Setup: I know the definition of a vector space: a set $$V$$ over a field $$F$$ such that is closed under vector addition and scalar multiplication.

My question: Is there a name for spaces that are closed under addition and scalar multiplication by positive real numbers? I am in particular thinking about spaces of square integrable functions on the real line (or some compact interval of it) that are everywhere greater than or equal to zero (call it $$S$$), with the property that it is closed under $$af_1+bf_2$$, where $$f_1,f_2\in S$$, and $$a,b\in\mathbb{R}_+$$. I know this is not a vector space (for example no $$f\in S$$ besides $$f=0$$ has an additive inverse), but is there another name for this space? I want to understand if this space (and others like it) has a notion of a "basis".