How $\mathbb{R}^{N}$ is a function space

Question

Let $\mathbb{R}^{N} = \{x_{1},x_{2},...,x_{N}\}, x_{i} \in \mathbb{R}$.

Definition of function space:

Let X be an arbitrary nonempty set and let E be a vector space. Denote by F the space of all functions from X into E. Then F becomes a vector space if the addition and multiplication by scalars are defined in the following natural way:

(f +g)(x) = f(x)+g(x),

(λf)(x) = λf(x)

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I wish to know how $\mathbb{R}^{N}$ is a function space.

Is the nonempty set X here is {1,2,...,N}?

What is the vector space E here?

Can you clarify?

Answer 1

Note that a point in $\mathbb R^N$ can be thought of as a choice of $N$ images in $\mathbb R$, one for each of the domain values (indices) $1,2,\ldots,N$. That is, it is a particular function $\{1,2,\ldots,N\}\to\mathbb R$.

So the vector space here is $\mathbb R$. Addition and scalar multiplication of functions (elements of $\mathbb R^N$) is defined in terms of the corresponding operations on the image values (elements of $\mathbb R$), just as you indicate.

(Technically, we should write $\mathbb R^{\{1,2,\ldots,N\}}$, but we can easily identify $N$ with the set $\{1,2,\ldots, N\}$ of size $N$ and write $\mathbb R^N$ instead.)