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Index

Abelian group, 38 absolute homogeneity, 198 absolutely convergent series, 201 absolutely convex set, 270 absolutely homogeneous functional,

198 absolutely homogeneous metric, 198 absolutely summable family, 344, 346 absolutely summable sequence, 201 absorbing set, 270 accumulation point, 116-118 additive Abelian group, 38 additive mapping, 55 additively invariant metric, 198 additivity, 313 adherent point, 115-117 adjoint, 379, 387-390,454 algebra, 82 algebra with identity, 83 algebraic complement, 68, 69, 71, 72,

74,289 algebraic conjugate, 56 algebraic dual, 56 algebraic linear transformation, 80

algebraic operator, 281 algebraically disjoint, 67 annihilator, 303 antisymmetric relation, 8 approximate eigenvalue, 453 approximate point spectrum, 452 approximation spectrum, 453 Arzela-Ascoli Theorem, 163 associative binary operation, 38 Axiom of Choice, 15

backward bilateral shift, 293, 423 backward unilateral shift, 248, 292,

421,474 Baire Category Theorem, 145-147 Baire metric, 187 Baire space, 147 balanced set, 270 Banach algebra, 222 Banach limit, 304 Banach space, 200, 209, 214, 219, 233,

271,295 Banach-Steinhaus Theorem, 242, 295 Banach-Tarski Lemma, 11

518 Index

barrel,271 barreled space, 271, 295 Bessel inequality, 355 best linear approximation, 332 bidual,266 bijective function, 5 bilateral ideal, 83 bilateral shift, 293, 422, 425 bilinear form, 312 bilinear functional, 312 binary operation, 37 block diagonal operator, 280 Bolzano-Weierstrass property, 156 Boolean sum, 4 boundary, 181 boundary point, 181 bounded above, 9, 223, 231 bounded away from zero, 225, 272 bounded below, 9, 223, 228, 231 bounded family, 208 bounded function, 9, 88, 216, 272 boundedinverse,223,228,229 bounded linear operator, 220 bounded linear transformation, 215,

216 boundedsequence,89,127,272 bounded set, 9,87, 152,270,272 bounded variation, 185 boundedly complete lattice, 10

C* -algebra, 396 canonical basis for .e~, 363 canonical basis for IFn , 54 canonical bilateral shift, 423, 470 canonical unilateral shift, 422, 470 Cantor set, 190 Cantor-Bernstein Theorem, 11, 17 cardinal number, 15 cardinality, 14-16, 18,21,22,31-35,

77 Cartesian decomposition, 429 Cartesian product, 4, 13,66, 167, 177,

189,193,206 Cauchy criterion, 127,274,344

Cauchy sequence, 127,134,185-187, 271

Cayley transform, 501 chain, 12 characteristic function, 16, 27 clopen set, 184 closed ball, 101 closed convex hull, 269 Closed Graph Theorem, 229 closed linear transformation, 287-289 closed map, 113 closed set, 113, 116, 128, 149, 150 Closed Set Theorem, 118 closed subspace, 179 closure, 113-115, 180, 184 cluster point, 116, 118 codimension, 70, 81 codomain, 5 coefficients, 276 cohyponormal operator, 446 coisometry, 391,422 collinear vectors, 409 comeagre set, 144 commensurable topologies, 106 commutant, 284 commutative algebra, 83 commutative binary operation, 38 commutative diagram, 6 commutative group, 38 commutative ring, 39 commuting operators, 281, 489, 492,

506 compact extension, 257 compact operator, 250-254, 256, 257,

301,302,308,428,434,436, 474-480,482-484,486,489, 495,503,504

compact restriction, 256 compact set, 148, 149, 158, 159,236,

237 compact space, 148, 149, 157-160,

193-195 compatible topology, 269 complementary linear manifolds, 341

complementary projection, 290 complementary subspaces, 289, 290,

339,342 complete lattice, 10, 11,45,176,212,

282 complete set, 271 complete space, 128, 130, 131, 133-

135,137-141,145-147,156, 158-160,185-189,191,192

completely continuous, 250 completion, 138-141,240-242,340 complex field, 40 complex linear space, 41 composition of functions, 6 compression spectrum, 452 condensation point, 180 conditionally compact, 149 cone, 80 conjugate space, 265 connected set, 184 connected space, 265 connectedness, 184 constant function, 5 continuity, 97, 106 continuity of inner product, 411 continuity of inversion, 298 continuity of metric, 177 continuity of norm, 200 continuity of scalar multiplication, 269 continuity of vector addition, 269 continuous composition, 103, 176 continuous extension, 135-138 Continuous Extension Theorem, 262 continuous function, 96-100,102,113,

122,149-151,159,182,216 continuous inverse, 223 Continuous Inverse Theorem, 228 continuous linear extension, 237-239 continuous linear transformation, 215 continuous projection, 221, 290,300 continuous spectrum, 451, 474 continuous restriction, 176 Continuum Hypothesis, 22 contraction, 97, 218, 298, 400, 409

Index 519

Contraction Mapping Theorem, 132 contrapositive proof, 3 convergence,93,106 convergence-preserving map, 100 convergent nets, 96 convergent sequence, 93-96,103,106,

127 convergent series, 201, 274 convex functional, 198 convex hull, 76, 269 convex linear combination, 76 convex set, 269 convex space, 271 coordinates, 49 coset, 43 countable set, 18 countably infinite set, 18 covering, 8, 148 cyclic subspace, 283 cyclic vector, 283

De Morgan laws, 4, 24 decomposition, 72-74, 342,406,429,

486,507 decreasing function, 10 decreasing increments, 185 decreasing sequence, 13 dense in itself, 126 dense linear manifold, 211, 237-239,

333 dense set, 121, 122, 145, 146, 179,

329 dense subspace, 122, 135-138 densely embedded, 138 densely intertwined, 283, 284 denumerable set, 18 derived set, 116, 118 diagonal mapping, 173, 183,218,289 diagonal operator, 218, 225, 246, 249,

254,255,299,302,400,416, 470,505

diagonal procedure, 22, 154, 162 diagonalizable operator, 416, 467,489,

504,505

520 Index

diameter, 87 dimension, 53, 77, 81, 359 direct proof, 2 direct sum, 66-68, 74, 75, 206-208,

220, 279, 280, 290, 321-323,325,337,338

direct sum decomposition, 68, 72, 74, 75,342

direct summand, 75, 279 directed downward, 10 directed set, 10 directed upward, 10 disconnected set, 184 disconnected space, 184 disconnection, 184 discrete dynamical system, 79 discrete metric, 105 discrete set, 126, 184 discrete space, 105 discrete topology, 105 disjoint linear manifolds, 67 disjoint sets, 4 disjointification, 29 distance, 86, 89 distributive laws, 39 division ring, 39 domain, 5 Dominated Extension Theorem, 262 doubleton, 4 dual space, 265

e-net, 151 eigenspace, 451, 485, 490, 501 eigenvalue, 451, 479, 484, 507 eigenvector, 454, 476, 505 embedding, 6 empty function, 28 empty sum, 349 equicontinuous, 160,295 equiconvergentsequences, 170, 186 equivalence, 230 equivalence class, 7 equivalence relation, 7 equivalent metrics, 106, 108, 176

equivalent norms, 231, 232 equivalent sets, 14 equivalent spaces, 231 Euclidean metric, 87 Euclidean norm, 203, 318 Euclidean space, 87,203,318 eventually constant, 108 eventually in, 103 expansion, 49, 276 extension by continuity, 258 extension of a function, 6 extension ordering, 28 extension over completion, 141, 142,

241,257,341

F -space, 271 Fa, 147 field,40 final space, 405 finite sequence, 13 finite set, 14 finite-dimensional space, 53, 61, 65,

77,232-237,244,251,267, 291,292,301,303,346,355, 382,424,466,504

finite-dimensional transformation, 79, 251

finite-rank transformation, 79, 251,254, 292,302

first category set, 143, 145, 147, 148 fixed point, 6, 11, 132 Fourier coefficients, 360 Fourier series expansion, 360 Fourier Series Theorem, 360 Fr6chet space, 271 Fredholm Alternative, 480 Fubini's Theorem, 390 Fuglede-Putnam Theorem, 492 full direct sum, 206 function, 5

Ga,147 Gelfand-Beurling formula, 460 Gelfand-Naimark Theorem, 396

Gram-Schmidt process, 357 graph, 4 greatest lower bound, 9 group, 38, 229

Hahn Interpolation Theorem, 178 Hahn-Banach Theorem, 260-262 Hamel basis, 48, 49, 51-53, 355, 358 Hausdorff Maximal Principle, 23 Hausdorff space, 179 Heine-Borel Theorem, 158 Hermitian operator, 396 Hermitian symmetric functional, 312 Hermitian symmetry, 313 Hilbert cube, 195 Hilbert space, 317 Hilbert-Schmidt operator, 435, 436 Holder conjugates, 164 Holder inequalities, 164-166 homeomorphic spaces, 108, 150 homeomorphism, 108, 111, 114, 147,

150 homogeneity, 313 homogeneous mapping, 55 hyperinvariant linear manifold, 284 hyperinvariant subspace, 284 hyperplane, 82 hyponormal operator, 446-448, 450,

499,506-508 hypo normal restriction, 501

ideal, 83 idempotent, 7, 25, 70, 301 identity element, 38, 39, 83 identity map, 6 identity operator, 222 image of a point, 5 image of a set, 5 inclusion map, 6 increasing function, 10, 11 increasing sequence, 13 index set, 12 indexed family, 12 indexing, 12

Index 521

indiscrete topology, 105 induced equivalence relation, 8 induced topology, 104, 199,315 induced uniform norm, 218 inductive set, 2 infimum, 9, 13 infinite diagonal matrix, 218 infinite sequence, 13 infinite set, 14 infinite-dimensional space, 53, 356,

359 initial segment, 13 initial space, 405 injection, 6 injective function, 5, 25, 26 injective linear transformation, 56 inner product, 312 inner product axioms, 312 inner product space, 313, 323 inner product space e~(X), 323 interior, 120 interior point, 121 intertwined operators, 283 intertwining transformation, 283 invariant linear manifold, 73-75, 280 invariant set, 6 invariant subspace, 280, 282-284, 392,

393,500-503,507 invariant subspace problem, 495, 507 inverse element, 38, 83 inverse image, 5 Inverse Mapping Theorem, 228 inverse of a function, 7, 25, 223 inversely induced topology, 179 invertible element of 8[X, y], 228 invertible function, 7, 26 invertible linear transformation, 58 invertible operator in 8[X], 229 involution, 26 isolated point, 125, 126, 142 isometric isomorphism, 239-242, 267,

268,292-294 isometrically equivalent operators, 294

522 Index

isometrically equivalent spaces, 110, 138, 140, 378

isometrically isomorphic spaces, 239, 241,266-268,304,338

isometry, 110, 195,239,292,294,298, 301,339,391,442,465,508

isomorphic equivalence, 64-ti6 isomorphic linear spaces, 59-ti3, 65-

67 isomorphism, 59-ti2, 64, 65, 67

Jensen inequalities, 166

kernel, 56 Kronecker delta, 54

lattice, 10,27,45,212,281,282 Laurent expansion, 460 Law of the Excluded Middle, 2 least-squares, 427 least upper bound, 9 left ideal, 83 left inverse, 25 limit, 29, 93, 96 limit inferior, 29, 169 limit superior, 29, 169 linear algebra, 82 linear basis, 48 linear combination, 46 linear composition, 78 linear dimension, 53, 357, 359 linear equivalence relation, 42 linear extension, 57, 258-262 linear functional, 55 linear manifold, 43, 209,325 linear restriction, 56, 78 linear space, 40 linear space .c[X, YJ, 56, 78 linear span, 45 linear topology, 269 linear transformation, 55, 57, 58, 62,

65 linear variety, 82 linearly independent set, 47, 352

linearly ordered set, 12 Liouville Theorem, 450 Lipschitz condition, 97 Lipschitz constant, 97 Lipschitzian mapping, 97,195 locally compact, 194 locally convex space, 271 Lomonosov Theorem, 495 lower bound, 9 lower limit, 169 lower semicontinuity, 175

map, 5 mapping, 5 Mathematical Induction, 2, 13 matrix, 65 maximal element, 9 maximal linear variety, 82 maximal orthonormal set, 353-355,

357 maximum, 9 meagre set, 143 metric, 85 metric axioms, 85 metric generated by a norm, 199,200 metric generated by a quasinorm, 271 metric space, 86 metrizable, 105 minimal element, 9 minimum, 9 Minkowski inequalities, 165 Mobius transformation, 501 modus ponens, 2 monotone function, 10 monotone sequence, 13 multiplication operator, 497 multiplicity, 421-426, 451 mutually orthogonal projections, 371

natural embedding, 267 natural isomorphism, 65, 67,341,342 natural projection, 222, 290 neighborhood, 101, 179 neighborhood base, 271

net, 14 neutral element, 38 nilpotent linear transformation, 80 nilpotent operator, 282 nondegenerate interval, 21 nondenumerable set, 18 nonmeagre set, 144 nonnegative contraction, 399,433 nonnegative functional, 198 nonnegative homogeneity, 198 nonnegative operator, 399-402, 406,

407,430,432-434,442,445, 463,499,505-507

nonnegative quadratic form, 312 nonnegativeness, 86, 199,313 nontrivial hyperinvariant subspace, 285 nontrivial invariant subspace, 281-284,

286 nontrivial linear manifold, 43 nontrivial projection, 70 nontrivial reducing subspace, 392,492 nontrivial ring, 39 nontrivial subset, 3 nontrivial subspace, 210 norm, 198,218 norm axioms, 198 norm induced by an inner product,

315,316 norm topology, 200, 315 normal operator, 441-448, 455, 480-

499,503-508 normal restriction, 501 normaloid operator, 397, 447, 463-

466,477 normed algebra, 222 normed linear space, 199 normed space, 199,222 normed space B[X, Y], 217, 218 normed spaces f~(X) and f~(X),

208 normed vector space, 199 nowhere continuous, 98 nowhere dense. 142. 143. 147 nuclear operator. 435

null function, 42 null space, 56, 216

Index 523

null transformation, 56, 217 nUllity, 79 numerical radius, 462-466 numerical range, 463

one-to-one correspondence, 5, 11 one-to-one mapping, 5 open ball, 100 open map, 108 Open Mapping Theorem, 225 open set, 10 1, 104, 105 open subspace, 179 operator, 220 operator algebra B[ X], 220, 222, 229,

246,253,284,307 operator convergence, 244-246, 249,

250,296-300,306-308,371, 372,381-385,401,403,418-420,422,424,428,430,462, 496,502

operator matrix, 280 operator norm property, 220 orbit, 282 order-preserving correspondence, 23 ordered n-tuples, 13 ordered pair, 4 ordering, 8 ordinal number, 23 origin of a linear space, 41 orthogonal complement, 328, 339, 342,

367 orthogonal dimension, 357, 359 orthogonal direct sum, 325, 338, 341,

342,367,392,416,421,469, 490

orthogonal family, 351, 352, 355, 356 Orthogonal Normalization Lemma, 415 orthogonal projection, 367-376, 398,

442,502,505-507 orthogonal projection onto M, 369,

392,415,485,502 orthogonal sequence, 324

524 Index

orthogonal set, 324, 352 Orthogonal Structure Theorem, 335 orthogonal subspaces, 327,328,335-

339,342,468,469,484,485, 487

orthogonality, 323 orthonormal basis, 354, 356, 360, 363-

365 orthonormal set, 352-354

p-integrable functions, 92 p-summable family, 207, 344, 348 p-summable sequence, 88 pair, 4 parallelogram law, 315 Parseval identity, 360 part of an operator, 444 partial isometry, 404-408 partial ordering, 8 partially ordered set, 8 partition, 8 perfect set, 126, 147 point of accumulation, 116-118 point of adherence, 115, 116 point of continuity, 97 point spectrum, 451 pointwise bounded, 160,242 pointwise convergence, 95, 243 pointwise totally bounded, 160 polar decomposition, 406, 408, 443 polarization identities, 315, 316 polynomial, 62, 80, 282 positive functional, 198 positive operator, 399, 400, 430, 431,

433,498,505 positive quadratic form, 312 positiveness, 86, 199,313 power bounded operator, 246 power of a function, 7 powersequence,246,282,300 power set, 4 pre-Hilbert space, 313 pre-image, 5 precompactness, 155

Principle of Contradiction, 2 Principle of Recursive Definition, 13 Principle of Superposition, 78 product metric, 167 product of cardinal numbers, 35 product space, 167, 177, 183, 189,

193 product topology, 193 projection, 70-74,82,221,300 projection on M, 71, 73, 74 projection operator, 221 Projection Theorem, 339, 342 proof by contradiction, 2 proof by induction, 2 proper subset, 3 proportional vectors, 409 pseudometric, 91 pseudometric space, 91 pseudonorm, 198,203 Pythagorean Theorem, 324, 351

quadratic form, 312 quasiaffine transform, 285 quasi affinity, 285 quasiinvertible transformation, 285 quasi nilpotent operator, 459,465,471,

508 quasinorm, 270 quasinormal operator, 443, 448, 498,

508 quasinormed space, 270 quasisimilar operators, 285 quasisimilarity, 286 quotient algebra, 83 quotient norm, 214 quotient space, 7, 42, 44, 69, 83,91,

139,204,205,213,214,240, 320,321

Radon-Nikod'yn Theorem, 491 range, 5 rank,79 rare set, 142 real field, 40

real linear space, 41 reducible operator, 392, 492 reducing subspace, 392, 484, 488, 492,

499,502 reflexive relation, 7 reflexive spaces, 267, 309 relation, 4 relative complement, 4 relative metric, 86 relative topology, 179 relatively closed, 179 relatively compact, 149, 159 relatively open, 179 residual set, 144, 146, 148 residual spectrum, 451 resolution of the identity, 371-376 resolvent function, 450 resolvent identity, 450 resolvent set, 449, 450 restriction of a function, 5 restriction of a function, 6 Riemann-Lebesgue Lemma, 426 Riesz Decomposition Theorem, 507 Riesz Lemma, 236 Riesz Representation Theorem, 376 right ideal, 83 right inverse, 25 ring, 39 ring with identity, 39

scalar, 40 scalar multiplication, 40 scalar operator, 219, 281 scalar product, 312 Schauder basis, 276, 287, 302 Schwarz inequality, 314 second category set, 144, 145 second dual, 266 self-adjoint operator, 396--401, 403,

429,430,432,442,456,465, 500-508

self-indexing, 12 semi-inner product, 320 semi-inner product space, 320

Index 525

semicontinuity, 175 seminorm, 198,203 seminormal operator, 446, 448 separable space, 123-126, 153, 183,

211,256,265,267,268,276, 291,294,357

sequence, 13 sequence of partial sums, 168,201 sequentially compact set, 155 sequentially compact space, 155-158 sesquilinear form, 312 sesquilinear functional, 312 set, 3 shift, 248, 292, 293, 421-424, 448,

468-471,474 similar linear transformations, 65, 80 similar operators, 286, 294 similarity, 65, 80, 286, 294 simply ordered set, 12 singleton, 4 span, 45,46,211 spanned linear manifold, 47 spanned subspace, 211 spanning set, 211 spectral decomposition, 486, 492 Spectral Mapping Theorem, 457 spectral measure, 490 spectral radius, 457-464, 473, 483,

499,501 Spectral Theorem, 484, 486, 490, 492 spectraloid operator, 464, 465 spectrum, 449-457, 466--469, 474-

480,499,502,506-508 spectrum diagram, 452 square root, 401 square root algorithm, 174 square-summable family, 344, 350, 351 square-summable net, 323 square-summable sequence, 322, 325 stability, 246-248,307,384,422,432,

461,497,499,503 strict contraction, 97, 218, 298 strictly decreasing function, 10 strictly decreasing sequence, 13

526 Index

strictly increasing function, 10 strictly increasing sequence, 13 strictly positive operator, 398, 400,

430,431,433,455,598,505, 507

strong convergence, 243, 245-248, 250, 296,299-301,371,372,376, 382,420,422,424,432,434, 490

strong limit, 244 stronger topology, 106 strongly bounded, 242 strongly closed, 249, 299 strongly stable operator, 246-248, 384,

422,432,434,497,503 subadditive functional, 198 subadditivity, 199 subalgebra, 83 subcovering, 148 sublattice, 10 sublinear functional, 198 subnormal operator, 444-446, 448, 508 subsequence, 14 subset, 3 subspace of a metric space, 86, 180 subspace of a normed space, 209,210,

234,326,329-331,335 subspace of a topological space, 179 Successive Approximation, 132 sum of cardinal numbers, 35 sum of linear manifolds, 44, 45, 67,

68 summable family, 343, 344, 346, 348,

350,351 summable sequence, 201, 274, 275,

279 sup-metric, 90, 91 sup-norm, 210 supremum, 9,13 surjective function,S, 25, 26 surjective isometry, 110, 138-142 symmetric difference, 4, 24 symmetric functional, 312 symmetric relation, 7

symmetry, 86

Tietze Extension Theorem, 178 Tikhonov Theorem, 193 topological base, 124 topological embedding, 109 topological invariant, 109, 147, 150,

183, 184 topological isomorphism, 231, 239,

290,291,294 topological linear space, 269 topological space, 105 topological sum, 213, 335 topological vector space, 269 topologically isomorphic spaces, 231,

235 topology, 104, 105 total set, 211 totally bounded, 152-156, 158, 159,

161-163 totally cyclic linear manifold, 283 totally disconnected, 184, 187 totally ordered set, 12 trace, 438 trace-class operator, 435, 436, 438,

439 transformation,S transitive relation, 7 triangle inequality, 86, 199 trichotomy law, 12 two-sided ideal, 83, 253

ultrametric, 186 ultrametric inequality, 186 unbounded linear transformation, 235,

289,366 unbounded set, 87 unconditionally convergent series, 348 unconditionally summable, 348, 350 uncountable set, 18 uncountably infinite set, 18 undecidable statement, 23 underlying set, 40, 86 Uniform Boundedness Principle, 242

uniform convergence, 244-246, 248, 273,296-300,381,420,422, 436

uniform homeomorphism, 109, 134, 137

uniform limit, 244 uniformly bounded, 242 uniformly closed, 249 uniformly continuous composition, 176 uniformly continuous function, 97, 134-

137,151,154,216 uniformlyequicontinuous, 161 uniformly equivalent metrics, 110, 177,

232 uniformly homeomorphic spaces, 110,

134, 155, 177 uniformly stable operator, 246, 248 unilateral shift, 292, 421-425, 448,

468 unit vector, 352 unital algebra, 83 unital algebra C[X], 56, 78, 83, 222 unital Banach algebra, 222 unital normed algebra, 222, 284 unitarily equi valent operators, 493, 494,

506 unitarily equivalent spaces, 342, 343 unitary operator, 423, 433, 442, 455,

498,500,505,506 unitary space, 87, 203 unitary transformation, 340, 343

upper bound, 9 upper limit, 169

Index 527

upper semicontinuity, 175 usual metrics, 86, 89, 93 usual norms, 202,203, 205, 208,218

value of a function,S vector, 40 vector addition, 40 vector space, 40 von Neumann expansion, 296

weak convergence, 306, 308, 309, 376-387,401,418,420,432

weak* convergence, 308 weak limit, 306 weaker topology, 106 weakly bounded, 410 weakly closed, 384 weakly closed convex cone B+[1"i],

400,430 weakly stable operator, 307,423,432,

503 Weierstrass Theorems, 124, 159 weighted bilateral shift, 470, 471 weighted sum of projections, 374,416,

481,482,488,504 weighted unilateral shift, 470 well-ordered set, 12

Zermelo Well-Ordering Principle, 23 Zorn's Lemma, 17