Nonarchimedean Functional Analysis by Peter Schneider
By Peter Schneider
The current ebook is a self-contained textual content which leads the reader via the entire vital elements of the speculation of in the neighborhood convex vector areas over nonarchimedean fields. you possibly can discover an expanding curiosity in tools from nonarchimedean useful research, rather in quantity idea and within the illustration concept of p-adic reductive teams. The ebook supplies a concise and transparent account of this concept, it conscientiously lays the rules and in addition develops the extra complicated issues. even if the booklet may be a beneficial reference paintings for specialists within the box, it's in most cases meant as a streamlined yet distinct creation for researchers and graduate scholars who desire to follow those equipment in several components.
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In particular, avn1 + cB ⊇ avn2 + c2 B ⊇ . . ⊇ avni + ci B ⊇ . . With B all the convex subsets avni + ci B are closed in V . It follows that the limit v in V of the sequence (avni )i lies in the intersection i avni + ci B. We consequently have pB (v − avni ) ≤ |c|i for any i ∈ IN. This shows that the sequence (avni )i converges to v already in VB . Our original Cauchy sequence (vn )n therefore converges in VB to a−1 v. 18: Suppose that V is Hausdorff and quasi-complete; then Ls (V, W ) and LB (V, W ), for any B-topology which is finer than the weak topology, have the same class of bounded subsets.
We now assume that the assertion ii. holds and we have to conclude from this that V is bornological. Let L ⊆ V be a lattice satisfying (bor). ii and our assumption it suffices to show that the gauge pL is bounded on any bounded subset B ⊆ V . But using a scalar a ∈ K such that B ⊆ aL we have pL (v) ≤ |a| for v ∈ B. Next we establish the equivalence of i. and iii. Again we first assume that V is bornological. It is trivial that a continuous linear map respects bounded subsets. To show that f is continuous provided it has this latter property it suffices to check that, given an open lattice M ⊆ W , its preimage f −1 (M ) satisfies the condition (bor).
Set U := ker(f ). Since the residue class projection W −→ W/U is open it suffices to show that the induced continuous linear map W/U −→ V is open. We claim that W/U satisfies the same assumption as W . Define Un := i−1 n (U ) ′ for any n ∈ IN. The induced linear maps in : Wn /Un −→ W/U are injective and continuous. The assumption that V is Hausdorff implies that each Un is closed in Wn . It therefore follows from Prop. 3 that each Wn /Un is a Fr´echet space. Obviously we have W/U = n i′n (Wn /Un ).