Summable family

The concept of summable family aims at extending calculations of nap S to the case of an infinite number of terms. Contrary to the concept of series, it is not supposed that the terms are given in the form of an ordered continuation. It is thus a question of being able to define the sum of it in a total way, without specifying the order in which one proceeds. So the sommability is a property more demanding than the convergence of series, and has additional properties.

The sommability gives in particular a useful framework for the study of the double series.

Preliminary example

That is to say the continuation of general term $u_n = \ frac {(- 1) ^n} n$ for strictly positive N whole. Is one can answer in several ways the question “which the sum of the terms of this continuation? ”

The theory of the series amounts successively summoning all the terms by forming the sum partial of order NR (nap of the NR first terms) and while passing in extreme cases. Here it is simpler to calculate

U_ {2N} = \ sum_ {n=1} ^ {2N} u_n = \ sum_ {p=1} ^N \ frac1 {2p} - \ sum_ {p=0} ^ {N-1} \ frac1 {2p+1}

2 \ sum_ {p1} ^N \ frac1 {2p} - \ sum_ {p1} ^ {2N} \ frac1 {p}

By the use of the Formule of Euler it is found that the continuation $U_ {2N}$ tends towards - ln (2), and one finds the same limit for the continuation $U_ {2N+1} =U_ {2N} - \ frac1 {2N+1}$ . One can thus affirm that the series converges and write

\ sum_ {n=1} ^ {+ \ infty} \ frac {(- 1) ^n} {N} = \ ln 2

However, by summoning the same terms in another way, it is possible to obtain a distinct result: one decides much more quickly to summon the even terms than odd the

V_N = \ sum_ {p=1} ^ {2N} \ frac1 {2p} - \ sum_ {p=0} ^ {N-1} \ frac1 {2p+1}

\ sum_ {p1} ^ {2N} \ frac1 {2p} + \ sum_ {p1} ^N \ frac1 {2p} - \ sum_ {p1} ^ {2N} \ frac1 {p}

\ frac12 \ ln 2 +o (1)

When NR tends towards the infinite one, $U_N$ and $V_N$ do not have the same limit, whereas these limits are obtained by summoning one and only once each term of the continuation! For such a continuation the order in which one proceeds to carry out the summation changes the result. For a more complete study of this situation to see the article Theorem of Riemann for the semi-convergent series.

One wishes to introduce a definition of the sum which excludes this kind from situation, and which ensures that the summation gives the same result whatever the selected order.

Definition of a summable family of realities or complexes

One can a priori define only the sum of a number finished of complex real numbers or . One gives oneself a Ensemble $I \,$ and a family $(u_i) _ {I \ in I} \,$ of real numbers. It is said that the family $(u_i) _ {I \ in I} \,$ is summable if there exists a reality $A \,$ such as:

\ forall \ varepsilon > 0, \ exists J_ {0} \ subset I, \, J_ {0} \, \ mathrm {finished}, \, \ forall J \ subset I, \, J \, \ mathrm {finished}, \, J_ {0} \ subset J \ Rightarrow \ left| \ sum_ {I \ in J} u_i - has \ right| \ Leq \ varepsilon

The real one has is single and is called nap of the family

(u_i) _ {I \ in I} \,

. It in general is noted

\ sum_ {I \ in I} u_ {I}

Of course if the unit $I \,$ itself is finished, the family is automatically summable, her sum having the accustomed value. The same if the family admits only one finished number of nonnull values (almost null Famille), it is summable and one finds the usual value of the sum.

The sommability has properties of linearity. The sum of two summable families is still a summable family, and the sums are added. The product of a summable family by a reality λ is a summable family, and it sum is multiplied by λ.

Remark

This writing resembles a kind of passage in extreme cases on increasingly large finished units. In fact, one can say indeed that it is about a limit according to a Base of filter.

Case of positive realities

If the $(u_ {I}) _ {I \ in I} \,$ are positive realities, one has a rather convenient characterization of the sommability, coming from what, for the finished units, the value of the sum is increasing for inclusion.

Thus the family is summable if and only if the unit

E = \ left \ {\ sum_ {I \ in J} u_ {I}, \, J \ subset I, \, J \, \ mathrm {finished} \, \ right \}

is raised. The sum of the family

(u_ {I}) _ {I \ in I} \,

is then the upper limit in

\ R \,

of the unit

E \,

One can then use results of comparison: if two families of positive realities $(u_ {I}) _ {I \ in I}, \, (v_ {I}) _ {I \ in I} \,$ admettent the same whole of indexing and check for all I , $u_i \ Leq v_i$ and if the family $(v_ {I}) _ {I \ in I} \,$ is summable, then the family $(u_ {I}) _ {I \ in I} \,$ is too.

Bond with the concept of series

One calls support of the family the unit J of the indices such correspondent in the nonnull terms. If a family of positive realities indexed by a unit I is summable then necessarily her support is finished or Dénombrable. This is why one is interested in general only in the sommability on countable units.

At least from the formal point of view, it is then possible to compare the sommability with the problems of convergence of series. That is to say $(u_ {I}) _ {I \ in \ NR} \,$ a countable family of positive realities. This family is summable if and only if the series $\ sum u_i$ converges, and in this case summons series and of the family are equal.

Families of realities or complexes

One can bring back the study of the summable families of realities or complexes to that of the families of positive realities. Indeed, one can prove that such a family $(u_ {I}) _ {I \ in I}$ is summable if and only if the $family (|u_ {I}|) _ {I \ in I}$ is summable.

In the case of a family of realities, one can introduce the positive Partie $a^ {+} = max (has, 0)$ and the negative part $a^ {-} = max (- has, 0)$ (these two numbers are positive) reality has . Then since $\ forall I \ in I, u_ {I} ^ {+} \ Leq |u_ {I}|$ , by comparison the family of positive realities $(u_ {I} ^ {+}) _ {I \ in I}$ is summable. The same applies to the family $(u_ {I} ^ {-}) _ {I \ in I} \,$ . It is shown whereas one with the following equality:

\ sum_ {I \ in I} u_ {I} = \ sum_ {I \ in I} u_ {I} ^ {+} - \ sum_ {I \ in I} u_ {I} ^ {-}

For a family of complex numbers one can in the same way separate the elements from the family in left real and imaginary. The family is summable if and only if the family of the real parts is summable and that of the imaginary parts also

\ sum_ {J \ in I} u_j = \ sum_ {J \ in I} \ Re (u_j) +i \ sum_ {J \ in I} \ Im (u_j)

In particular, a family $(u_ {N}) _ {N \ in \ NR} \,$ of realities or complexes is summable if and only if the series $(u_ {N}) _ {N \ in \ NR} \,$ is absolutely convergent.

Associativeness and commutation

One considers again a family $(u_i) _ {I \ in I}$ of real numbers or complexes.

The order of the terms is not taken into account in the definition of the sommability. Thus if σ is a Permutation of the unit I , then the families $(u_i) _ {I \ in I}$ and $(u_ {\ sigma (I)}) _ {I \ in I}$ is of comparable nature, and if they are summable, have the same amount. This property is the generalization of the commutation finished sums.

To generalize the associativeness it is necessary to introduce a partition $(I_t) _ {T \ in T}$ of the unit I . There is then equivalence between these two properties

the family $(u_i) _ {I \ in I}$ is summable;
for all T of T , the family $(u_i) _ {I \ in I_t}$ is summable, of nap $S_t$ , and the family of the $(S_t) _ {T \ in T}$ is it also summable.

Moreover in this case there is equality of the sums

\ sum_ {I \ in I} u_i = \ sum_ {T \ in T} \ left (\ sum _ {i_t \ in I_t} u_ {i_t} \ right)

Families of vectors

One considers a real or complex vector space E provided with a Norme $\|. \|$ . That is to say $(u_i) _ {I \ in I}$ a family of elements of E (vectors). This family is known as summable when there exists a vector has E such as

\ forall \ varepsilon > 0, \ exists J_ {0} \ subset I, \, J_ {0} \, \ mathrm {finished}, \, \ forall J \ subset I, \, J \, \ mathrm {finished}, \, J_ {0} \ subset J \ Rightarrow \ left \| \ sum_ {I \ in J} u_i - has \ right \| \ Leq \ varepsilon

Obviously, it is a direct generalization of the concept of summable family of scalars. The element has is called the nap of the family $(u_i) _ {I \ in I}$ . It is only defined by the property above. Again the whole of the summable families on E constitutes a vector space, the application which with a family associates her sum being linear.

The near total of the properties of the summable families of scalars can be wide to the summable families of vectors, provided that the vector space normalized E is a Espace of Banach.

Criterion of Cauchy in spaces of Banach

The criterion of Cauchy is in general a requirement of sommability, but within the framework of spaces of Banach, it provides a requirement and sufficient from which the remarkable properties associated with the sommability rise.

A family $(u_i) _ {I \ in I}$ satisfies the criterion of Cauchy when

\ forall \ varepsilon >0, \ exists J \, \ mathrm {finished} \, \ forall K \, \ mathrm {finished} \, \ qquad (J \ course K) = \ emptyset \ Rightarrow \ left \|\ sum_ {K \ in K} u_k \ right \|< \ varepsilon

In picturesque terms J contains almost all the sum since with what is elsewhere one does not manage to exceed $\ epsilon$ .

It is clear that any summable family checks the criterion of Cauchy. As it was already known as, it is proven that the reciprocal one is true when E is complete.

Using the criterion of Cauchy, one can establish that, in a space of Banach, any subfamily of a summable family is summable. One can as show as the property of associativeness is still valid. But the most interesting application is the possibility of introducing a new criterion of sommability: absolute sommability.

Absolute sommability

The family $(u_i) _ {I \ in I}$ is known as absolutely summable when the family of positive realities $(\|u_i \|) _ {I \ in I}$ is summable.

The criterion of Cauchy has as a corollary which any absolutely summable family is summable, and checks the triangular Inégalité wide

\ left \|\ sum_ {I \ in I} u_i \ right \|\ Leq \ sum_ {I \ in I} \|u_i \|

It was already seen that for the families of realities or complexes, sommability and absolute sommability were equivalent. If E is of finished size equivalence is still true, but it is not the case in general.

Against example

One takes for E space

\ ell^2

of the continuations of summable Carré. It is indeed a space of Banach. The family considered is

(U_n) _ {N \ in \ N^*}

with for all N , U_n the continuation of which all the terms are null except that of order N which is worth

\ frac1n

. There is thus

\ left \|U_n \ right \|= \ frac1n

. The family of these standards is thus not summable. On the other hand the family

(U_n) _ {N \ in \ N^*}

it even is summable, of sum the continuation

U= (\ frac1n) _ {N \ in \ N^*}

which belongs well to

\ ell^2

Sommability and linear form

That is to say a second vector space normalized $(F, \|. \|)$ and a linear application continues $\ lambda: (E, \|. \|) \ rightarrow (F, \|. \|)$ . For any summable family $(u_i) _ {I \ in I}$ of vectors of $E$ , the $family (\ lambda (u_i))_ {I \ in I}$ is summable. Moreover, one a:

\ sum_ {I \ in I} \ lambda (u_i) = \ lambda \ left I} u_i \ right

From this property one can develop the weak concept of summation . A family of vectors $(u_i) _ {I \ in I}$ is known as slightly summable when, for all linear form continues $\ lambda$ E , the family of realities $(\ lambda (u_i))_ {I \ in I}$ . The preceding property is reformulated as follows: a summable family is slightly summable.

The weak sommability takes all its direction when dual topological the $' E'$ separates the points from E . If such is the case, one calls nap of the family slightly summable $(u_i) _ {I \ in I}$ the single element there of E , checking for any linear form $\ lambda$ , the identity: $\ lambda (there) = \ sum_ {I \ in I} \ lambda (u_i)$ . One poses: $y= \ sum_ {I \ in I} u_i$ .

Implication in finished dimension

In finished dimension the choice of the standard does not have importance: all the standards on a real or complex vector space in finished dimension are equivalent. Let us indicate by K the body R or C and consider a family $(u_i) _ {I \ in I}$ of vectors of $K^n$ . For the vector $u_i$ , one notes the components $(u_i^1, \ dowries, u_i^n)$ . The family $(u_i) _ {I \ in I}$ is summable if and only if the families of the components $(u_i^1) _ {I \ in I}$ ,…, $(u_i^n)$ are summable.

The direct direction is a simple reformulation of the preceding property, the reciprocal direction on the other hand request to be written but does not present truly any difficulty.

Product in the algebras of Banach

If $(has,|. |)$ is an algebra of Banach, it is legitimate to wonder how behaves the product with respect to the summation. If $(u_i) _ {I \ in I}$ and $(v_j) _ {J \ in J}$ is two summable families, then the family produced $(u_iv_j) _ {I, J \ in I \ times J}$ is summable and one a:

\ left I} u_i \ right. \ left J} v_j \ right= \ sum_ {(I, J) \ in I \ times J} u_iv_j

This property can reinterpret using the double series.

Complementary references

Internal bonds

Série (mathematics)
Série doubles

Publications

Laurent Schwartz, mathematical Methods for physical sciences , Paris, Hermann, 1965,2e edition, chapter 1

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