# Is the periodic function a vector space?

• golmschenk
In summary, the conversation discusses whether the set of all periodic functions of period 1 is a vector space or not. The conversation includes the definition of a vector space and a specific function that raises questions about its status as a subspace. The conversation then concludes that the set is not a vector space due to the function not meeting the criteria of a subspace.

## 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.

golmschenk said:
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?

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

## 1. Is the set of periodic functions a vector space?

Yes, the set of periodic functions is a vector space. This means that it satisfies all the properties of a vector space, such as closure under addition and scalar multiplication, and contains a zero vector.

## 2. What are the characteristics of a vector space?

A vector space must have a set of vectors, a field of scalars, and operations of addition and scalar multiplication defined on it. It must also satisfy certain properties, such as associativity, commutativity, and distributivity.

## 3. Can a non-periodic function be considered a vector in the space of periodic functions?

No, a non-periodic function cannot be considered a vector in the space of periodic functions. This is because the space of periodic functions only includes functions that have a repeating pattern, while non-periodic functions do not have this property.

## 4. How does a periodic function behave under scalar multiplication?

Under scalar multiplication, a periodic function will be stretched or compressed along the x-axis by a factor of the scalar. This will not affect the periodicity of the function.

## 5. Can a periodic function be considered a subspace of a larger vector space?

Yes, a periodic function can be considered a subspace of a larger vector space, such as the space of all continuous functions. This is because it satisfies all the properties of a vector space and can be defined as a subset of a larger space.