I have a problem in linear algebra course, and I'm looking to solve it by myself, but I'm confused with notation since my teacher never mention it in class. It says:
Let $S$ be a nonempty set and $F$
is a field. Prove that for any $s_{0}\in S$
, $\left\{ f\in\mathcal{F}\left(S,F\right)\colon f\left(s_{0}\right)=0\right\}$
is a subspace of $\mathcal{F}\left(S,F\right)$
I'm confused with the $\mathcal{F} \left(S,F\right)$ notation. It certainly a vector space (or so I thought), but which vector space? Is that a standard notation?
If this information gives a better clue, it comes from Linear Algebra text by Stephen H. Friedberg, Arnold J. Insel, and Lawrence E. Space.
(I can't check the book myself because I copy the questions from my friend's note)
Best Answer
Define vector space operations on $\mathcal{F}(S,F)$, functions mapping set $S$ to field $F$ as vector addition:
$$ (f+g)(s) = f(s) + g(s) $$
and scalar multiplication:
$$ (a*f)(s) = a*f(s) $$
for any functions $f,g:S \rightarrow F $ and scalar $a \in F$.