## 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 $$\mathbb K,X=\mathbb R$$?