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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)

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Most likely, the set of functions $S \to F$. – Zhen Lin Sep 3 '13 at 14:25
Or more likely/specifically the set of linear maps $S \to F$. – Anthony Carapetis Sep 3 '13 at 14:26
@Anthony: $S$ is just a set, not necessarily a vector space. – arjafi Sep 3 '13 at 14:27
Oh right, never mind me. It indeed works with $S$ any set, no vector space structure on $S$ is necessary. – Anthony Carapetis Sep 3 '13 at 14:28
up vote 5 down vote accepted

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 $a \in F$.

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