Question

Let 𝑆 be any non-empty set and let 𝑉 (𝑆) be the set of all real valued functions on ℝ. Define addition on 𝑉 (𝑠) by (𝑓 + 𝑔)(𝑥) = 𝑓 (𝑥) + 𝑔(𝑥) and scalar multiplication by (𝛼 ⋅ 𝑓 )(𝑥) = 𝛼𝑓 (𝑥). Check that (𝑉 (𝑆), +, ⋅) is a vector space.

Answer 1

Recall the definition of the vector space:

A vector space over $\mathbb{C}$ is a set $V$ with operations of addition:$V\times V\rightarrow V$ and scalar multiplication: ${\mathbb{C}}\times V\rightarrow V$ satisfying the following properties:

1. Commutativity: $f+g=g+f$ for all $f,g\in V$.
2. Associativity: $(f+g)+h=f+(g+h)$ for all $f,g,h\in V$.
3. Additive identity: there exists $0\in V$: $0+f=f$ for all $f\in V$.
4. Additive inverse: For every $f\in V$ there exists $g\in V$ satisfying: $f+g=0$.
5. Multiplicative identity: there is $1f=f$ for all $f\in V$.
6. Distributivity: $a(f+g)=af+ag$ and $(a+b)f=af+bf$ for all $a,b\in{\mathbb{C}}$ and all $f\in V$.

We will check the properties for $V(S)$. Properties $1$ and $2$ are satisfied due to the properties of standard addition and definition: $(f+g)(x)=f(x)+g(x)$. Consider zero function . It satisfies property $3$. For any function $f(x)$ consider the function $g(x)=-f(x)$. It satisfies property $4$ . Properties $5$ and $6$ are satisfied due to the definition $(\alpha\cdot f)(x)=\alpha f(x)$ and properties of standard multiplication.