Example

Tag 05JR, permanent. Cite it as [Bourbaki, Tag 05JR].

Algebra, chapter II, § 1. Modules

(4) A field is both a left and a right vector space with respect to any of its subfields.

(5) The real number space of three dimensions $\mathbf{R}^3$ is a vector space with respect to the field of real numbers $\mathbf{R}$ , the product $tx$ of a real number $t$ and a point $x$ with coordinates $x_1, x_2, x_3$ being the point with coordinates $tx_1, tx_2, tx_3$ . Similarly, the set of real-valued functions defined on an arbitrary set F is a vector space with respect to $\mathbf{R}$ , the product $tf$ of a real number $t$ and a function $f$ being the real-valued function $x \mapsto tf(x)$ .*

For two families $(x_i)_{i \in I}, (y_i)_{i \in I}$ of elements of an A-module E of finite support (I, § 2, no. 1), the following equations hold:

(1)

\sum_{i \in I} (x_i + y_i) = \sum_{i \in I} x_i + \sum_{i \in I} y_i

(2)

\alpha \cdot \sum_{i \in I} x_i = \sum_{i \in I} (\alpha x_i) \quad \text{for all } \alpha \in A;

the equations are immediately reduced to the analogous equations for finite sums by considering a finite subset H of I containing the supports of

(x_i)

and

(y_i)

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

I, § 2, no. 1