What Constitutes a Vector Space in Function Sets?

[stupidmonkey]

Homework Statement

Hello, this is not a problem but I want some clarification with the following paragraph (starting my linear algebra course!):

Let S be any nonempty set and F be any field*, and let F(S,F) denote the
set of all functions from S to F. Two functions f and g in F(S, F) are called
equal if f(s) = g(s) for each s in S. The set F(S,F) is a vector space with
the operations of addition and scalar multiplication defined for f, g in F(S, F)
and c in F by
(f + g)(s)=f(s)+g(s) and (cf)(s)=c[f(s)]
for each s in S. Note that these are the familiar operations of addition and
scalar multiplication for functions used in algebra and calculus.

*my prof said we can assume the field is R, all real numbers

Homework Equations

The Attempt at a Solution

What does it mean "from S to F"?
Does it mean that any function in F(S,F) can take in any values of S and produce a value in F?

I'm also confused about notation that my professor uses.
R^X to denote the set {f, X->R} where R is all reals and X is some non-empty set of numbers. Does this also mean the same thing as F(X,R)?

 

[Mark44]

Homework Statement

Hello, this is not a problem but I want some clarification with the following paragraph (starting my linear algebra course!):

Let S be any nonempty set and F be any field*, and let F(S,F) denote the
set of all functions from S to F. Two functions f and g in F(S, F) are called
equal if f(s) = g(s) for each s in S. The set F(S,F) is a vector space with
the operations of addition and scalar multiplication defined for f, g in F(S, F)
and c in F by
(f + g)(s)=f(s)+g(s) and (cf)(s)=c[f(s)]
for each s in S. Note that these are the familiar operations of addition and
scalar multiplication for functions used in algebra and calculus.

*my prof said we can assume the field is R, all real numbers

Homework Equations

The Attempt at a Solution

What does it mean "from S to F"?
Does it mean that any function in F(S,F) can take in any values of S and produce a value in F?

Yes.

I'm also confused about notation that my professor uses.
R^X to denote the set {f, X->R} where R is all reals and X is some non-empty set of numbers. Does this also mean the same thing as F(X,R)?

Yes.