Thread: Vector Space Proof

Let V denote the set of all differentiable real valued funtion defined on the real line. Prove that V is a VS under the operation of addition and scalar multiplication defined as follows:

addition: (f+g)(s)=f(s)+g(s), and multiplcation: (cf)(s)=c[f(s)]

I just don't know how to get started with this problem. I guess I'm having a problem thinking...what function are they talking about? How can you generalize all trig function, exponential function, polynomial into this? I don't know how to show the 8 axioms in this example. Like with the polynomials I know that i could define something like : anx^n+(An-1)X^n-1....AnX^1+An and then work with this, but I don't know how to proceed in the above.

Any help would be much appreciated.

Originally Posted by jusstjoe

Let V denote the set of all differentiable real valued funtion defined on the real line. Prove that V is a VS under the operation of addition and scalar multiplication defined as follows.

Is the derivative of a sum of two differentiable functions the sum of their derivatives?

Is the derivative of a scalar times a differentiable function that scalar times that function's derivative?