Is the periodic function a vector space?

[golmschenk]

Homework Statement

Is the following a vector space:
The set of all periodic functions of period 1? ( i.e. f(x+1)=f(x) )

Homework Equations

If v1 and v2 are in V then v1 + v2 is in V

If v1 is in V then c*v1 is in V where c is a scalar

The Attempt at a Solution

I'm thinking no because a function like:
f(x) = 2*x mod 2
If you multiply by 1/2 you have
f(x) = x mod 2

What I'm wondering is if the mod 2 is effected by the scalar multiplication of 1/2. Is it or is this not a subspace? Thanks for your time.

 

[foxjwill]

f(x) = 2*x mod 2
If you multiply by 1/2 you have
f(x) = x mod 2

Let's check this using $x=17$:

$$\frac{(2\cdot5)\mod 2}{2} = \frac{\mathop{\mathrm{remainder}}(10,2)}{2} = \frac{0}{2} = 0,$$​

while

$$5\mod 2 = \mathop{\mathrm{remainder}}(5,2) = 1.$$​

Evidently, the two expressions are not the same. Can you see why this is?

 

[golmschenk]

Right. It's a scalar multiple of the entire function. Got it. Thanks!