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A-antisymmetric set 170 Banach-Dieudonne theorem 181 absolute basis 314, 510 - disk 151 - summability 305, 341 - lattice 413 absolutely continuous operator 477 - -Mackey theorem 151 - (r-), huH 101, 102 - -Mazur distance 416, 462 - (p-)summing operator 428, 467 -- theorem 173 absorbent sequence 253 -, (r-)-ideal 425 - set 32 - -Schauder theorem 96 abstract kernel theorem 487, 499 - space 116 adjoint map, operator 160, 246 - -Steinhaus theorem 220 a. e. bounded function 25 ~-approximation property 398 affine map 489 barrel 150, 219 Alaoglu-Bourbaki theorem 157 barreHed, No- 251 algebra 170 -, C o- 249 algebraic dual 20 - les (topology) 219 alm ost shrinking basis 304 -, /00- 249 - weak*-topology 178 barycenter 489 AL-space 414 basic sequence 296 AM-space 414 basis in a linear space 19 approximable operator 419 --- 510 -, p-- operator 440 --- tvs 292 approximation numbers 439 -,0- 31 - property 301, 399 - of a 153 - theorem for linear forms 177 --- cone 489 d-space 479 ~-complete les 195 associated Hausdorff tvs 77 B-completeness, B,-completeness 183 - Schwartz space (topology) 203 ~-convergence 195 - (strongly) nuclear space (topology) Bessel's inequality 316 508 bidual 132, 222 - (ultra-)bornological space (topology) 22, 145 276 - map 22, 88, 220 Auerbach basis 291 bi orthogonal sequence 289 Auerbach's lemma 291 bipolar 148 - theorem 149 Baire algebra 138 Bishop's theorem 170 - category 87 Bochner integral 340, 341, 392 - measure 138 Borel algebra 138 -set 138 - measure 138 - space, tvs 87 - set 138 544 Index bornivorous sequence 253 continuous convergence 174 - set 222 - A.-- measure 138 bornological, (,qj-) -les 272 c~-nuc1ear space 491 bornology 152, 153 convergence,,qj- 195 bounded function 25, 26 -, ~- 199 - 223, 271, 372 - space 193 - set 34,44 convex function 489 boundedly complete basis 302 -, (r-)-set, hull 101,102 box topology 75, 111 copy of aspace 310 ,qj-topology 44, 152 coset 16 countable additivity 137 canonical bilinear form 146 - compactness 64, 189 cartesian product 17 counting measure 26, category (first, second) 87 Cauchy filter, net, sequence 57, 60 Db-space 270 Cebysev polynomials 320 DCF-space 366 chain of O-neighbourhoods 33 defining family (F-) 39, 108 Choquet simplex 489 c5-ultrafilter 282 circ1ed set, hull 32 dj-space 257 c1osed, '}"--set 178 D F-space 257 - graph theorem 92, 221, 277 diagonal operator 377 - ideal 419 diameter 207 - linear map 92 diametral dimension 209, 485 - range theorem 185 dimension, linear 20 c10sure of an ideal 419 directed family (F-seminorms) 39 co-d-space 493 direct sum, linear 17 codimension 20 -, locally (r-)convex 110 coefficient functionals 292 -, topological 78 compact-open topology 45 disk 151 - operator 368, 419, 452 distinguished les 280 compactology 157 distributions 86 compatible topology 149 dual, algebraic 20 - web 89 - ideal 420 complementation 18, 77 - of a direct sum 168 complete, ,qj--les 195 -- a product 165 - set, tvs 57, 60 --- quotient 164 - topological ideal 424 --- subspace 163 completeness, B-, Br- 183 -- 1. (p) 239 completing web 90 - paiiing 145 completion 61 -, topological 35 complex measure 137 Dunford-Pettis-Phillips theorem 394 component of an ideal 419 -- property 327, 472 concave function 489 Dvoretzky-Rogers theorem 434 cone 489 Dynin-Mitiagin theorem 510 conjugate number 48 consistent topology 149 Eberlein's theorem 189 continuity, y- 176 ~-convergence 199 - theorem for bases 293 B-norm 345 Index 545

B-product 344 Grothendieck's inequality 458 B-topology 344 equicontinuous basis 296 Haar system 300 - compactology 157 Hahn-Banach extension theorem 126, - convergence 199 127, 132 - set 156 - deeomposition 138 - weak*-topology 178 Hamel basis 19 equivalence of ideal quasi-norms 423 Helly's lemma 165 -- inductive systems 82 hermitean form 241 -- projective systems 38 Hermite functions 318 essentially bounded function 25 Hilbert-Schmidt operator 457, 468 evaluable lcs 222 - space 243 evaluation map 132, 222 hilbertian (semi-)norm 243 expansion operators 292 hilbertisable lcs 243 extension lemma 29 Hölder's inequality 48 - of uniformly continuous maps 61 holomorphic vector valued function - property 132 362 - theorems 126, 127, 132 homomorphism 77, 184 extremal, (r-)- set 133 HS-operator 457, 468 extreme, (r-)- point 133 hyperplane 20, 22 hypocontinuity 342 factorization theorem for p-summing operators 433 ideal of operators 418 --- weakly compact operators 374 - -quasinorm 422 filter, 0- 31 - -(r-)norm 423 finest linear topology 76, 83, 123 - -topology 424 -lc, topology 111,123 identity map 18 finite bornology 153 inductive limit 82, 110 - operator 330, 419 - linear topology 74 Fischer-Riesz theorem 66 - lc,-topology 110 F-norm 38 - sequence 84 Frechet space 109 - system 82 F- 38 infra-barrelled lcs (topology) 222 fully complete operator 420 --Ptäk space 194 function space 25 - -Schwartz space 479 fundamental sequence 257 injective Banach space 340 - hull of an ideal 421 - ideal 421 Gantmacher's theorem 373 - tensor norm 345 gauge functional 105 -- product 344 -, (r-) 105 integral bilinear form 383 gDF-space 257 - operator 383, 419, 434 Gram-Schmidt isometrie map 42 orthonormalization 317 isomorphism 18 graph 92 Grothendieck constant 459 joint continuity 88 --Pietsch criterion 497 Jordan decomposition 138 Grothendieck's completenes theorem 176 kernel 18, 499 546 Index kernel theorem 487, 500 Minkowski's inequality 49 Khintchine's inequality 318 mixed topology 270 Körnura theorem 500 monotone F-norm 41 Köthe sequence space 27 230 - set 27 multiplication theorems for ideals 445, Krein-Milman theorem 134 471 - -Smulian theorem 184 Krein's theorem 192 Nachbin-Shirota theorems 234, 284 kacspace 70 natural factorization 18 nearly open map 186 Laguerre functions 318 neighbourhood, 0- 31 le, le" les, le, s 109 norm,6- 345 Legendre functions 318 -, F- 38 lifting property 119 -, n- 326 limit space 193, 218 -, (r-) 105 linear combination 16 normable, (r-)- tvs 115 - dimension 20 normed, (r-)- ideal 423 - form 20,22 -, (r-)- space 115 - hull 16 nowhere dense 87 - independence 19 nuclear bilinear form 487 - map 17 -, tff-- sequence 500 - space 15 - operator 376, 419 - span 16 -, p-- operator 434, 439, 467 - subspace 16 - space 345, 479, 482 - topology 30 null sequence 34 local completeness 196, 198 - space 18 - convergence 196 localization theorem for absorbent open map 76, 184 sequences 255 - mapping theorem 95, 221 --- strictly webbed tvs 97 operator 17, 418 locally bounded tvs 114 - (s-)norm 118 - (r-)convex space (topology) 108, 109 Orlicz-Pettis theorem 308 -, (J-- topological space 270 orthogonal complement 244 tj,-sequence 425 - projection 245 ~-space 462 - system 315 - vectors 244 Mackey-Arens theorem 158 orthonormal basis 316 - completeness 196 - sytem 315 - convergence 196 orthonormalization 317 - topology 155 - -Ulam theorem 282 parallelogramm identity 241 Markushevich basis 289 - inequality 238 Mazur's theorem 130 Parseval's identity 316 meagre 87 periodic function 52 metric approximation property 408 Pettis integral 367 metrizable ideal 424 n-norm 326 - tvs 40 n-topology 324 minimal tvs 302 polar 148 Index 547 positive 139 regular inductive limit (system) 83 - measure 137 - measure 138 power series space 211 replete space 284 precompact operator 368 representable operator 392 - set 64 Riesz representation theorem 139 pre-Hilbert space 243 principle of local reflexivity 389 saturated bomology 153 -- uniform boundedness 220 scalar product 241 probability measure 432 Schatten-von Neumann classes 454 probleme de topologies 342 Schauder basis 292 product, (;- 344 Schauder's theorem 369 projection 18, 77, 245 Schur's lemma 218 projective limit 37, 42, 63 Schwartz space (topology) 201,479 - representation 43, 63, 117 Schwarz' inequality 241 - system 37 segment, (r-) 133 - tensor norm 326 self-adjoint subalgebra 170 -- product 324 semi-Montel space 229 -- topology 324 seminorm, F- 38 - topology 35 -, (r-) 105 Prokhorov's theorem 268 semi-I'/-reflexive les 280 property DN 503 - -reflexive les 227 pseudoconvex tvs (topology) 109 separate continuity 88 P-space 71 separation of points 146, 171 Ptitk space 194 - theorems 130, 131 Pythagoras' theorem 244 sequence space 27 sequential compactness 64, 191 quasi-No-barrelled 251 - completeness 57 - -barrelIed les (topology) 222 sequentially, b-- closed linear map 277 - -co-barrelled 249 - closed linear map 92 - -complete tvs 57 -- set 59 - -too -barrelIed 249 - separable 300 - -normable les 214 series 58 - -normed ideal 423 shrinkable set 106 - -p- 437 shrinking basis 302 Q-space 284 signed measure 137 quotient space 17, 76 Silva space 270, 416 simple function 68, 338 Rademacher functions 318 - topology 44 Radon-Nikodym property 194, 394 shortest distance 240 range (space) 18 Smulian's theorem 191 rapidly g;J-convergent sequence 273 Sobczyk's theorem 160 - decreasing 28, 51 space,.s:1- 479 rare 87 -, Sp- 430, 486 realeompact 284 - with a bomology 218 reduced inductive limit (system) 82 --- compactology 193 - projective limit (system) 38 --- mixed topology 270 reflexive, 1'/-- les 280 spectral theorem for compact operators -les 227 451 548 Index standard unit vectors 68 translation invariant metric 38 Stone-Weierstrass theorem 171 transposed map 160 strict inductive limit (sequence) 84 triangle, (r-)- inequality 38, 105 - topology 46 trigonometric system 300 - web 96 tvs 30 strictly b-sequentially closed linear map type of apower series space 211 277 - convex Banach space 248 Ulam measure 282 - webbed tvs 96 ultrabornological lcs 272 strong topology 154 ultrafilter, b- 282 strongly inaccessible cardinal 282 unconditional basis 309 - nuclear operator 443 - summ ability 305, 316, 357 -- space 479, 504 uniform complementation 430 subalgebra 170 - continuity 56 sublinear functional 125 - convergence 44 subseries summability 307 - convexity 236 subspace . 16 uniformly tight 268 summing, p-- operator 428, 467 unit ball 115 support 46, 143 universal generators 205,206,500,507, surjective hull of an ideal 421 509 - ideal 421 valued, IK-- measure 137 tensor product 23 variation of a measure 137 --, injective (e) 344 vector measure 309 --, projective (n) 324 - space 15 topological basis 292 - complement 77 - dual 35 Walsh functions 318 - ideal 424 weak (Schauder) basis 295 - 30 - basis theorem 296 topologically free 289, 292, 490 - compactness 189, 205 topology, 14- 44, 152 - integral 362 -, e- 344 - ~-sequence 426 - of bi-equicontinuous convergence - topology 146 344 - *-topology 148 -- compact convergence 45 weakly 372, 419 -- pointwise convergence 44 - holomorphic function 362 -- uniform convergence 44 - integrable function 362 - on a finite-dimensional space 66 - precompact operator 372 -, n- 324 web 89 total subset 153 webbed tvs 90 totally bounded set 64 trace functional 386 Zorn's lemma 19 Mathematische Leitfäden Herausgegeben von em. o. Prof. Dr. phi!. Dr. h.c. mult. G. Köthe, Universität Frankfurt/M., und o. Prof. Dr. rer. nato G. Trautmann, Universität Kaiserslautern

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