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Set of functions

  1. Sep 7, 2012 #1
    1. The problem statement, all variables and given/known data

    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
    2. Relevant equations



    3. 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.
    RX 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)?
     
  2. jcsd
  3. Sep 7, 2012 #2

    Mark44

    Staff: Mentor

    Yes.
    Yes.
     
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