By Florian Cajori

ISBN-10: 0486621847

ISBN-13: 9780486621845

ISBN-10: 1418165557

ISBN-13: 9781418165550

Initially released in 1904. This quantity from the Cornell collage Library's print collections used to be scanned on an APT BookScan and switched over to JPG 2000 structure through Kirtas applied sciences. All titles scanned disguise to hide and pages could comprise marks notations and different marginalia found in the unique quantity.

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**Extra info for An introduction to the theory of equations**

**Example text**

For the moment, we pause in these theoretical developments to give some examples of sequences which are σ (V , V ∗ ) convergent but not · V convergent. 1. Take V = l 2 . An element v of V is a sequence of real numbers, v = (vk )k∈N , such that k∈N |vk |2 < +∞. The scalar product u, v := k∈N uk vk and the corresponding norm v = ( |vk |2 )1/2 give to V a Hilbert space structure. Consider the sequence e1 , e2 , . . , en , . . with en = (δn,k )k∈N , where δn,k (the Kronecker symbol) takes the value 1 if k = n and 0 elsewhere.

There is a natural partial ordering on the topologies on a given space X, which is induced by the inclusion ordering on the subsets of P (X): we will say that a topology τ1 is coarser or weaker than a topology τ2 and we write τ1 < τ2 if θτ1 ⊂ θτ2 , that is, if any element G of θτ1 also belongs to θτ2 . Conversely, we will say that τ2 is stronger or finer than τ1 . 1. The family of the topologies on a set X forms a complete lattice for the relation τ1 < τ2 (τ1 weaker than τ2 ), that is, given an arbitrary collection of topologies (τi )i∈I on X, (a) There exists a lower bound, that is a topology which is the largest among all the topologies weaker than the τi , i ∈ I .

Suppose that for each i ∈ I , a function fi : X → Yi is given. We want to investigate the topologies on X with respect to which all the functions fi are continuous, and, among these topologies, examine the question of the existence of a smallest (weakest) one. Noticing that for each i ∈ I , fi−1 (θτi ) := {fi−1 (Gi ) : Gi ∈ θτi } still satisfies the axioms of the open sets, we denote fi−1 (τi ) the corresponding topology on X, which is the weakest making fi (for i fixed) continuous. 2 that the answer to the previous question is given by τ = ∨i∈I fi−1 (τi ), whose precise description is given in the following.

### An introduction to the theory of equations by Florian Cajori

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