Help with vector space of real value functions

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Homework Statement


The set of all real-valued functions f defined everywhere on the real line and such that f(4) = 0, with the operations (f+g)(x)=f(x)+g(x) and (kf)(x)=kf(x)
verify if all axioms hold true.


Homework Equations


Axioms 1 and 6 These closure axioms require that if we add two functions that are defined at each x in the interval , then sums and scalar multiples of those functions are also defined at each x in the interval.

Axiom 4 This axiom requires that there exists a function 0 in , which when added to any other function f in produces f back again as the result. The function, whose value at every point x in the interval is zero, has this property. Geometrically, the graph of the function 0 is the line that coincides with the x-axis.

Axiom 5 This axiom requires that for each function fin there exists a function —f in , which when added to f produces the function 0. The function defined by has this property. The graph of can be obtained by reflecting the graph of f about the x-axis

Axioms 2,3,7,8,9,10 The validity of each of these axioms follows from properties of real numbers. For example, if f and g are functions in , then Axiom 2 requires that . This follows from the computation
(f+g)(x)=f(x)+g(x)=g(x)+f(x)=(g+f)(x)

The Attempt at a Solution


I couldn't find a good example in my textbook so I kinda don't know how to start this problem. Help is much appreciated.
 

Answers and Replies

  • #2
LCKurtz
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I don't know which numbers are which axioms but I will give you a hint on getting started. To show closure under addition you have to start with f and g in your set and show f + g is in your set. Obviously if f and g are functions on R then so is f+g. The only question is whether (f+g)(4) = 0. Does it? Etc.
 
  • #3
Thanks for trying to help LCKurtz. I still don't quite understand it though.
(f+g)(4)=f(4)+g(4)=0? I don't know what to use for f or g equation wise so I don't know how I'm suppose to plug in 4.
 
  • #4
LCKurtz
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Thanks for trying to help LCKurtz. I still don't quite understand it though.
(f+g)(4)=f(4)+g(4)=0? I don't know what to use for f or g equation wise so I don't know how I'm suppose to plug in 4.

Your set of functions are any real valued functions on R with the property that f(4) = 0. So if you start with f and g in your set, you are given that f(4)=0 and g(4)= 0 because that is what your set is. If you want to know if their sum is in your set you have to check if the sum's value at 4 is 0.
 

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