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Cambridge University Press 978-0-521-19243-9 - Locally Convex Spaces over Non-Archimedean Valued Fields C. Perez-Garcia and W. H. Schikhof Index More information

Index

absolutely convex, 83 Bessaga–Pelczynski Selection Principle, 234 absolutely convex hull, 84 best approximation, 5 absorbing, 85 bilinear form, 446 accumulation point, 446 bilinear map, 374, 446 adjoint, 272 Bipolar Theorem, 215 admissible topology, 235 boundary, 446 affine transformation, 445 bounded away from 0, 443 algebraic, 444 bounded map, 448 algebraic base, 445 bounded sequence, 112, 448 algebraic closure, 444 bounded set, 112, 448 algebraic dual, 445 bounded weak topology, 336 algebraically closed, 444 α-regular inductive sequence, 411 C1-function, 70 almost Hahn–Banach property, 199 Cn-function, 72 analytic element, 65 C∞-function, 72 A-normed space, 203 c-countably generated, 90 Approximation Theorem for linear forms, 243 c-compact, 154, 206, 244 Ascoli Theorem, 143 canonical norm on Kn, 20 associated norm, 38 Cauchy net, 100, 449 associated topologies, 336 Cauchy sequence, 449 automorphism, 444 Cauchy–Schwarz inequality, 38 centre of a ball, 1 Baire Category Theorem, 104, 448 characteristic, 444 Baire space, 448 characteristic function, 446 Banach disk, 414 clopen, 447 Banach space (over K), 15 closed, 446 Banach’s Open Mapping Theorem, 21 closed ball, 1 Banach–Dieudonne´ Theorem, 440 Closed Graph Theorem, 22, 111, 413 Banach–Steinhaus Theorem, 22, 111, 256 closed linear hull, 449 Banach–Stone Theorem, 79 closed unit ball, 6, 15 Banaschewski compactification, 55 closed unit disk, 6 barrel, 255 closely related, 101 barrelled, 255 closure, 446 barycenter, 240 cluster point, 448 base, 31, 338, 442, 446 codimension, 445 base of continuous seminorms, 94 coefficient functional, 338

468

© in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-19243-9 - Locally Convex Spaces over Non-Archimedean Valued Fields C. Perez-Garcia and W. H. Schikhof Index More information

Index 469

commutation relation, 80 Eberlein-Smulian˘ Theorem, 241 compact, 447 edged, 84, 88 compact open topology, 115 edged hull, 84 compact operator, 334 embedded, 447 compactifying operator, 309 embedding, 17, 97, 444, 447 compactoid inductive limit, 424 entire function, 62 compactoid inductive sequence, 424 equicontinuous, 449 compactoid operator, 302 equivalent norms, 18 compactoid regular inductive sequence, equivalent seminorms, 92 441 equivalent valuations, 6 compactoid set, 143 ε-dense, 91 complement, 24, 105, 445 ε-topology, 335 complemented, 24, 105 extreme point, 206 complete, 100, 449 extreme set, 206 completed tensor product, 395 completion, 100, 449 fenced disk, 63 complexification, 404 field extension, 444 connected, 447 field homomorphisms, 444 Continuity Lemma, 384 filter, 442 continuous, 447 finite rank, 445 continuously differentiable, 70 finite type, 182, 218 convergence of filters, 447 Frechet´ space, 108 convergence of nets, 447 free simplex, 207 convex, 87 Full Perturbation Lemma, 33 convex filter, 246 functionals, 445 convex hull, 88 fundamental sequence coordinate map, 338 fundamental sequence of bounded sets, 295 countable type, 28 fundamental sequence of compact sets, 116 fundamental sequence of complete bounded Decomposition Lemma, 236, 242 subsets, 126 degree, 444 fundamental sequence of equicontinuous sets, dense, 446 295 dense valuation, 7 derivative, 70 gauge, 85 diameter, 2 germs of analytic functions, 125 Dieudonne-Schwartz´ Theorem, 412 Goldstine Theorem, 276 differentiable function, 70 graph, 443 dimension, 445 Grothendieck, 413 direct sum, 445 Grothendieck–Kothe¨ counterexample, 432 directed set, 443 discrete topological space, 446 Hahn–Banach property, 199 discrete topology, 446 Hahn–Banach Theorem, 171, 173, 176, 178, discrete valuation, 7 193, 205, 217 disk, 62 Hasse derivatives, 131 distance between sets, 2 Hausdorff, 446 distributions, 371, 438 hemicompact, 116 divisible group, 444 Hilbert-like space, 40 dual, 97 homeomorphic, 447 dual of an inductive sequence, 415 homeomorphism, 447 dual space, 17, 97 homomorphism, 444 dual-separating, 202, 210, 211 hyperplane, 445

470 Index

ideal, 444 Mackey Convergence Condition, 426 immediate valued field extension, 11 Mackey space, 235 indiscrete topology, 446 Mackey topology, 235 inductive limit, 107 Mahler base, 79 inductive sequence, 107 maximal convex filter, 246 inductive system, 107 maximal ideal, 444 inductive topology, 121 maximal valued field, 11 inherited topology, 446 measure, 51 inner product, 38 measures, 263 interior, 446 mercury drop behaviour, 3 invariant metric, 108, 449 metric, 1 invariant subspace, 445 metric space, 1 inverse, 444 metrizable, 447 isolated, 446 metrizable locally convex space, 108 isometrical isomorphism, 448 Minkowski function, 85 isometrically embedded, 448 Montel, 312 isometrically isomorphic, 17, 448 multiplicative group of a field, 7 isometry, 448 isomorphic, 97, 444, 445 N-compact, 448 isomorphism, 444, 445 n-th derivative, 70 isosceles triangle principle, 3 n-th difference quotient, 71 n-th Hasse derivative, 72 K-algebra, 446 natural map, 8 k0-space, 117 neighbourhood, 446 Kothe¨ dual, 350 neighbourhood base, 446 Kothe¨ sequence space associated to a nested collection, 442 matrix, 356 net, 442 Kothe¨ space, 349 non-Archimedean valuation, 6 Krein–Milman compactoids, 206 non-Archimedean valued field, 6 Krein–Milman Theorem, 206 non-measurable cardinality, 443 Kronecker delta, 446 norm, 15 Krull valuation, 13 normable, 96 normal topology, 351 Lagrange polynomials, 187 normed direct sum, 21 (LF)-space, 435 normed product, 21 limit, 447 normed space (over K), 15 linear hull, 445 nuclear, 312, 320 linear manifold, 445 null sequence, 449 linearly homeomorphic, 17 Liouville Theorem, 64 of countable type, 175 Lipschitz, 448 of finite type, 218 Lipschitz constant, 448 (O.P.)-space, 240 Lipschitz norm, 16 open, 446 (LM)-space, 435 open ball, 1 local convergence, 305 open map, 447 locally compact, 448 Open Mapping Theorem, 21, 111, 308 locally constant, 47 open unit ball, 6, 15 locally convex direct sum, open unit disk, 6 101 operator, 302 locally convex direct sum topology, 101 Orlicz–Pettis space, 240 locally convex space, 93 orthocomplemented, 24 locally convex topology, 93 orthogonal, 23

Index 471

orthogonal base, 29 projective tensor product, 384 “orthogonal” base, 338 property (∗), 240 orthogonal complement, 24 pseudocompact, 267 orthogonal indexed system, 25 Pull Back Principle, 155, 156, 323 orthogonal projection, 24 orthogonal sequence, 25 quasicomplete, 113 “orthogonal” sequence, 162 quotient, 448 orthogonal set, 24 quotient map, 445, 448 orthogonal system, 24 quotient norm, 17 “orthogonal” system, 26 quotient seminorm, 98 orthogonal with respect to p, 162 quotient space, 17, 445 orthonormal base, 29 quotient topology, 17, 98, 448 orthonormal system, 24 orthoprojection, 24 radius, 1 overconvergent power series, 125 Radon measures, 437 rapidly decreasing at infinity, 143 p-adic complex number, 10 rapidly decreasing sequences, 371 p-adic Fourier Transform, 438 reflexive, 275 p-adic integer, 9 regular inductive limit, 411 p-adic Laplace Transform, 438 regular inductive sequence, 411 p-adic metric, 2 relatively closed, 447 p-adic number, 9 relatively compact, 448 p-adic valuation, 8 relatively open, 447 p-open, 92 residue class field, 7 p-topology, 92 resolution, 435 partition, 442 restriction topology, 446 perfect sequence space, 350 Riesz Representation Theorem, 52 Perturbation Lemma, 25, 33, 42 Riesz Theorem, 145 ϕ-null set, 263 rigid analytic function, 126 π-topology, 335 ring, 442 pointwise bounded, 256, 449 pointwise convergence, 114, 449 saturated, 93 polar inductive limit, 259, 422, 438 Schauder base, 31, 163 polar inductive sequence, 438 Schwartz space, 334 polar seminorm, 193 semi-metric, 2 polar set, 196 semi-Montel, 311 polar space, 193 semicompact inductive sequence, 439 polar topology, 193 semicompact operator, 334 polar topology associated to τ, 208 seminorm, 83 polarly barrelled, 256 seminorm of countable type, 175 power series, 62 seminorm of finite type, 218 precompact, 143, 449 semireflexive, 275 prime field, 444 separable, 28, 447 principal, 442 separate the points, 443 Principle of van Rooij, 23 separation of convex sets, 204, 240 Product Lemma, 305 sequence space, 125, 350 product of locally convex spaces, 99 sequentially closed, 447 product topology, 448 sequentially complete, 114, 449 projection, 24, 105, 445 sequentially continuous, 447 projective limit, 106 sequentially retractive inductive sequence, 441 projective sequence, 106 sides of a hyperplane, 205 projective system, 106 σ -compact, 448

472 Index

simple convergence, 114 totally disconnected, 447 small, 443 translate, 446 solid norm, 19 translation, 445 solid seminorm, 83 transseparable, 175 spherical completion, 11 trivial metric, 3 spherically complete, 4 trivial valuation, 7 splitting lemma, 389, 392 twist map, 375 step function, 47 Tychonov Theorem, 448 steps, 107 Stone–Cech˘ compactification, 55 ultrafilter, 442 strict inductive limit, 411 ultrametric, 2 strict inductive sequence, 411 ultrametric space, 2 strict topology, 119 ultrametrizable, 447 strictly differentiable, 70 unconditional summation, 16 strictly of countable type, 176 Uniform Boundedness Principle, 22, 111, 256 strong topology, 272 uniform convergence, 449 strong triangle inequality, 2, 6 uniform convergence on bounded sets, 272, stronger topology, 446 390 strongest locally convex topology, 103, 113, uniform convergence on compact sets, 115, 195, 216, 220, 319 396 strongly polar, 201 uniform convergence on compact sets of subbase, 446 difference quotients, 128 subring, 444 uniform convergence on compactoids, 315 subspace, 447 uniformizing element, 7 sum, 16, 41 uniformly continuous, 448, 449 sum of subspaces, 445 union, 446 summable, 16, 41, 449 unit disk, 6 support, 449 unit sphere, 6 support of a measure, 263 unit vectors, 29 upper half plane, 127 t-frame, 166 t-frame sequence, 166 valuation, 5 t-orthogonal, 27 value group, 7, 13 t-orthogonal base, 29 valued field, 5 t-orthogonal indexed system, 30 valued field extension, 7, 56 t-orthogonal sequence, 30 vanish at infinity, 48, 119 t-orthogonal set, 27 vanishing at infinity, 139 t-orthogonal with respect to p, 162 Volume Function, 165 t-orthogonal system, 27 Taylor formula, 72 W-topology, 118, 133 tensor product, 374 weak Hahn–Banach property, 199, 212 tensor product of seminorms, 378 weak star topology, 272 Tietze–Urysohn Extension Lemma, 49, 222 weak topology, 182, 211 topological base, 31 weaker topology, 446 topological field, 5 weakly compact, 232 topological space, 446 weakly convergent sequence, 228 topological vector space, 15, 91 weakly precompact, 232 topology, 446 weakly strict, 438 topology of countable type, 175 Weierstrass Theorem, 11, 190 topology of countable type associated to τ, 207 weight functions, 118 topology of finite type, 218 topology of finite type associated to τ, 209 zero-dimensional, 447