By Professor André Weil (auth.)

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**Example text**

1, choose the sequence W I , ... , Wn so that it inc1udes Wand W'. Then th. 1 gives us a basis {w I, ... , wn} of V which generates L as an R-module and contains bases for Wand for W'; renumbering this basis in an obvious manner, we may assume that {WI""'Ws } is a basis for Wand that {wr+I, ... ,ws} is one for W'. Call W" the subspace of V with the basis {W I , ... , Wr}. By prop. 5, M is open in W; therefore, by corollary 2 of prop. 6, we have M = M' + W', where M' = Mn W" is a K-Iattice in W".

Let L be a K -lattice in a left vector-space V of dimension n over K. Then there is a basis {v1, ... ,vn} of V such that L= Ißv i. Moreover, if W1 = V, W2 , ••• , w,. is any sequence of subspaces of V such that W; is a subspace of W;-l of codimension 1 for 2:::; i:::; n, the Vi may be so chosen that, for each i, {Vi' ... ' vn} is a basis of W;. Take a K-norm N such that L is defined by N(v):::; 1. Choose subspaces V1, ... , v" of V as in prop. 3 of § 1; then L= 2)L n V;). 7 to V; and L nV; for each i, we get the basis (v;).

Be any sequence with the limit °in K. Then the series LXi is commutatively convergent in K. ° For each nE N, put Rn =SUPi>nmodK(X i), Our assumption means that lim Rn = 0. Let now S, S' be two finite sums of terms in the series LXi' both containing the terms XO,XI, ... , X n and Locally compact fields 14 possibly some others. The ultrametrie inequality gives modK(S - S'):::;; Sn' The eonclusion follows from this at onee (the "filter" of finite sums of the series is a "Cauehy filter" for the distanee-funetion modK(x - y)).