The space of mappings

Let \(X\) be an arbitrary non-empty set and let \((\mathbb K,+,\cdot)\) be a field.

Denote \(\mathbb K^X\) the set of all mappings \(f: X\to \mathbb K\).

Define the sum \(\oplus\) in the \(\mathbb K^X\) and the multiplication \(\odot:\mathbb K\times \mathbb K^X \to \mathbb K^X\) as follows:

\[(f \oplus g)(x)=f(x)+g(x),\qquad\qquad (a \odot f)(x)=a\cdot f(x).\]

Variant 1
Show that \((\mathbb K^X,\oplus,\odot)\) is a vector space.
Resolution
Let \(\mathbf 0\) be the constant mapping to \(0\) (\(\forall x\in X\colon\mathbf 0(x) = 0\)).

Define the oposite mapping as \(\ominus f(x) = -f(x)\).

Now for example:

(SO) \((f \oplus \mathbf 0)(x) = f(x) + \mathbf 0(x) = f(x) + 0 = f(x)\),

(SI) \((f \oplus (\ominus f))(x) = f(x) + (-f(x)) = 0\) which holds for any \(x\in X\) and thus \((f \oplus (\ominus f)) = \mathbf 0\).
Variant 2
Which vector space we get if \(X\) is finite?
Result
For \(|X|=n\) we get the arithmetic vector space isomorphic to \(\mathbb K^n\).
Variant 3
Which vector space we get if \(X=\mathbb N\)?
Result
For \(X=\mathbb N\) we get the vector space of all sequences in \(\mathbb K\).
Variant 4
Which vector space we get if \(\mathbb K,X=\mathbb R\)?
Result
For \(\mathbb K,X=\mathbb R\) we get the vector space of all real functions.