§ 3. Infinite sums in the groups $\mathbf{R}^n$

3. Infinite sums in the groups $\mathbf{R}^n$

1. Summable families in $\mathbf{R}^n$

Since every point of $\mathbf{R}^n$ has a countable fundamental system of neighbourhoods, a family $(x_t)$ of points of the additive group $\mathbf{R}^n$ is summable only if the set of indices $t$ such that $X_t \neq 0$ is countable (Chapter III, § 5, no. 2, Corollary to Proposition 1); hence, essentially, the study of summable families in $\mathbf{R}^n$ is reduced to that of summable sequences. Nevertheless, for the same reasons as were given in Chapter IV, , in connection with summable families in $\mathbf{R}$ , we shall not impose any restriction, in what follows, on the cardinal of the index set

Proposition 1

A family* $(x_i)_{i \in I}$ of points $x_i = (x_{i,k})_{1 \leq k \leq n}$ of $\mathbf{R}^n$ is summable if and only if each of the $n$ families $(x_{i,k})_{i \in I}$ of real numbers is summable in $\mathbf{R}$ .

This follows from Chapter III, § 5, no. 4, Proposition 4.

The condition of Proposition 1 may be transformed as follows:

Theorem 1

A family* $(x_i)_{i \in I}$ of points of $\mathbf{R}^n$ is summable if and only if the family $(||x_i||)$ of Euclidean norms of the $x_i$ is summable in $\mathbf{R}$ .

This follows without trouble from Proposition 1, the condition for summability of a family of real numbers (Chapter IV, § 7, no. 2, Theorem 3), the inequalities

\sup_{1 \leq k \leq n} |x_{i,k}| \leq ||x_i|| \leq \sum_{i=1}^n |x_{i,i}|,

and the comparison principle (Chapter IV, § 7, no. 1, Theorem 2).

One can also proceed somewhat differently, by first establishing the following proposition:

Proposition 2

If* $(x_i)_{i \in I}$ is any finite family of points in $\mathbf{R}^n$ , *then

\sum_{i \in I} ||x_i|| \leq 2n \cdot \sup_{J \subset I} \left| \sum_{i \in J} x_i \right|.

For if $x_i = (x_{ij})_{1 \leq j \leq n}$ , we have $||x_i|| \leq \sum_{j=1}^n |x_{ij}|$ , hence

\sum_{i \in I} ||x_i|| \leq \sum_{j=1}^n \left( \sum_{i \in I} |x_{ij}| \right).

Now $\sum_{i \in I} |x_{ij}| = \sum_{i \in I} x_{ij}^+ + \sum_{i \in I} x_{ij}^-$ , and since for every subset $J$ of $I$ we have

-\sum_{i \in I} x_{ij}^- \leq -\sum_{i \in J} x_{ij}^- \leq \sum_{i \in J} x_{ij}^+ \leq \sum_{i \in I} x_{ij}^+

it follows that

\sum_{i \in I} |x_{ij}| \leq 2 \cdot \sup_{J \subset I} \left| \sum_{i \in J} x_{ij} \right|.

But $\left| \sum_{i \in J} x_{ij} \right| \leq \left| \sum_{i \in J} x_{ij} \right|$ , hence the inequality (1).

Now, Theorem 1 is equivalent to the following proposition (since $\mathbf{R}^n$ is a complete group): the family $(x_i)$ satisfies Cauchy’s criterion (Chapter III, § 5, no. 2, Theorem 1) if and only if the family $(||x_i||)$ also satisfies Cauchy’s criterion. Now the triangle inequality shows that this condition is sufficient, and the inequality (1) shows that it is necessary.

Furthermore we have the inequality

\left| \sum_i x_i \right| \leq \sum_i ||x_i||

which comes by passing to the limit from the analogous inequality for finite partial sums.

Corollary

A family* $(x_i)$ of points of $\mathbf{R}^n$ is summable if and only if the set of finite partial sums of the family is bounded in $\mathbf{R}^n$ .

By Theorem 1 and the triangle inequality, this condition is necessary; it is sufficient by the inequality (1) and Theorem 1.

Proposition 3

Let* $(x_\lambda)_{\lambda \in L}$ be a summable family of points of $\mathbf{R}^m$ , $(y_\mu)_{\mu \in M}$ a summable family of points of $\mathbf{R}^n$ , and let $f$ be a bilinear mapping of $\mathbf{R}^m \times \mathbf{R}^n$ into $\mathbf{R}^p$ . Then the family $(f(x_\lambda, y_\mu))_{(\lambda, \mu) \in L \times M}$ *is summable and we have

\sum_{(\lambda, \mu) \in L \times M} f(x_\lambda, y_\mu) = f \left( \sum_{\lambda \in \mu} x_\lambda, \sum_{\mu \in M} y_\mu \right).

To show that the family $(f(x_\lambda, y_\mu))$ is summable, it is sufficient by Proposition 1 to establish that each of the $p$ families formed by the coordinates of the points $f(x_\lambda, y_\mu)$ in $\mathbf{R}^n$ is summable: in other words, we can restrict ourselves to the case where $f$ is a bilinear form; but for such a form $f$ we have

f(x, y) = \sum_{i, j} a_{ij} x_i y_j,

and therefore we are brought back to the case $f(x, y) = x_i y_j$ , and in this case the result has already been proved (Chapter IV, § 7, no. 3, Proposition 1).

By specializing the function $f$ we obtain in particular the following corollaries:

Corollary 1

If* $(a_\lambda)_{\lambda \in L}$ is a summable family of real numbers and if $(x_\mu)_{\mu \in M}$ is a summable family of points of $\mathbf{R}^n$ , then the family $(a_\lambda x_\mu)_{(\lambda, \mu) \in L \times M}$ *is summable and we have

\sum_{(\lambda, \mu) \in L \times M} a_\lambda x_\mu = \left( \sum_{\lambda \in L} a_\lambda \right) \left( \sum_{\mu \in M} x_\mu \right).

Corollary 2

If* $(x_\lambda)_{\lambda \in L}$ and $(y_\mu)_{\mu \in M}$ are two summable families of points of $\mathbf{R}^n$ , then the family $(x_\lambda | y_\mu)$ (cf. Chapter VI, § 2, no. 2) is summable in $\mathbf{R}$ , *and we have

\sum_{(\lambda, \mu) \in L \times M} (x_\lambda | y_\mu) = \left( \sum_{\lambda \in L} x_\lambda \middle| \sum_{\mu \in M} y_\mu \right).

2. Series in $\mathbf{R}^n$

A series whose general term is $x_m = (x_{mi})_{1 \leq i \leq n}$ converges in $\mathbf{R}^n$ if and only if each of the $n$ series $(x_{mi})_{m \in \mathbf{N}}$ converges in $\mathbf{R}$ .

Definition 1

A series of points of* $\mathbf{R}^n$ *is said to be absolutely convergent if the series of Euclidean norms of its terms is convergent.

Proposition 4

A series of points of* $\mathbf{R}^n$ *is commutatively convergent if and only if it is absolutely convergent.

This is a consequence of Proposition 9 of Chapter III, § 5, no. 7 and Theorem 1 above.

The examples given in Chapter IV, § 7 show that a series in $\mathbf{R}^n$ can be convergent without being absolutely convergent.

Exercises

See the exercises for § 3.