Continuous Functions, Vector Spaces

[psholtz]

Homework Statement

Is the set of all continuous functions (defined on say, the interval (a,b) of the real line) a vector space?

Homework Equations

None.

The Attempt at a Solution

I'm inclined to say "yes", since if I have two continuous functions, say, f and g, then their sum f+g is also continuous.

But as per Fourier analysis, one can "arrive" at discontinuous functions by taking an infinite series of continuous functions.

Soo.. I'm not 100% certain.

 

[╔(σ_σ)╝]

I believe it is.
I don't think infinite sums are "allowed" in the definition; if they were then there would be convergence issues and certain things that ought to be vector spaces would not be.

I believe the definition require the sum of a finite linear combination to be in the vector space.