Corollary 2

Tag 0BRT, permanent. Cite it as [Bourbaki, Tag 0BRT].

General Topology, chapter IV, § 5. Real-valued functions

Let $f_1$ and $f_2$ be two real-valued functions defined on sets $X_1, X_2$ respectively; then if $A_1, A_2$ are any non-empty subsets of $X_1, X_2$ respectively, we have (24)

\sup_{(x_1, x_2) \in A_1 \times A_2} (f_1(x_1) + f_2(x_2)) = \sup_{x_1 \in A_1} f_1(x_1) + \sup_{x_2 \in A_2} f_2(x_2)

whenever both sides are defined; and if $f_1, f_2$ are $\geq 0$ in $A_1, A_2$ respectively we have

(25) \quad \sup_{(x_1, x_2) \in A_1 \times A_2} (f_1(x_1)f_2(x_2)) = \sup_{x_1 \in A_1} f_1(x_1) \sup_{x_2 \in A_2} f_2(x_2),

whenever both sides are defined.

This is an immediate consequence of the preceding corollary and Proposition 9 of no. 4.

In particular, if $A$ and $B$ are two subsets of $\overline{\mathbf{R}}$ such that the set $A + B$ of sums $x + y \ (x \in A, y \in B)$ is defined, we have

(26) \quad \sup (A + B) = \sup A + \sup B

if the right-hand side is defined. Again, if $A$ and $B$ are two subsets of $[0, + \infty]$ , we have

(27) \quad \sup AB = \sup A \cdot \sup B

whenever both sides are defined.

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

Cited by

Nothing.

Cites

Proposition 9