## The space of mappings

### Task number: 2505

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.

#### Variant 2

Which vector space we get if \(X\) is finite?

#### Variant 3

Which vector space we get if \(X=\mathbb N\)?

#### Variant 4

Which vector space we get if \(\mathbb K,X=\mathbb R\)?