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Notation

Part I

∅, x ∈ A, x ∈ A, A ⊂ B, 10N, N0, Z, Q, R, C, Z+, R+, 10A ∪B, A ∩B, A = B, 10P(X), 10Ac, 11f |A, 13∼, 14min, max, inf, sup, 15lim inf, lim sup, 16∏

j∈J Xj , 18card(A), 20|A|, 20d(x, y), 26Br(x), Bd(x, r), 26dp, d2, d∞, 27C([a, b]), B([a, b]), 27d(A,B), 28(xk)∞k=1, 29

limk→∞

xk = p, xk → p, xkd−−−−→

k→∞p, 29

τ , τ∗, 31τd, τ(d), 32τA, 33τ1 ⊗ τ2, 34intd(A), extd(A), ∂d(A), 35Vτ (x), V(x), 36NEFIS(X), 72K, 79Kn, 80V X , 80span(S), 80

L(V,W ), L(V ), 82Ker(A), Im(A), 82σ(A), 82x �→ ‖x‖X , ‖x‖, 92C(K), 93BX(x, r), B(x, r), 94‖A‖op, 94L(X, Y ), L(X), 94V ′, L(V, K), 94LC(X, Y ), LC(X), 95Γ(f), 99〈x, y〉, 103x⊥y, M⊥N , 103PM , PM (x), 105M⊥, M ⊕M⊥, 106M1 ⊕M2,

⊕j∈J Hj , 106

S1, Tr(A), 111S2, 〈A,B〉S2 , 112‖A‖HS , 112a⊗ b, X1 ⊗ · · · ⊗Xr, 83A⊗B, 84X ⊗π Y , X⊗πY , 90X ⊗ε Y , X⊗εY , 91∑

j∈J aj , 115m∗, 116M(ψ), 118Σ(τ), 119μ-a.e., 137f ∼μ g, 137f+, f−, 141∫

f dμ, 143Lp(μ), L∞(μ), ‖f‖Lp(μ), 152

694 Notation

ν+, ν−, |ν|, 159μ⊥λ, 1611A, 191[A,B], 192σA(x), 192A ∼= B, Hom(A,B), 195Spec(A), 208ρ(x), 205R(λ), 204Gel, Gel−1, 216H(Ω), 217

Part II

Spaces, sets

Z+, N0, Z, R, C, 298U|W , 330τX , (X, τX), 330(uj)j∈Z+ , 330L(X, Y ), L(X, Y ), 330supp, 239Ker, Im, 381Rn/Zn, 300Cm

1 , C∞(Tn), 300S1, 299Tn = Rn/Zn, 299L2, L2(Tn), 302TrigPol(Tn), 305Hs(Tn), 307Hs,t(Tn × Tn), 309C∞(Tn × Tn), 309C∞(Tn × Zn), 338Sm(Tn × Zn), Sm

ρ,δ(Tn × Zn),

S∞ρ,δ(Tn × Zn), S−∞(Tn), 338

Sm(Rn × Rn), 260Sm

ρ,δ(Rn × Rn), 261

Op(Sm(Tn × Zn)),Op(Sm

ρ,δ(Tn × Zn)), 338

Op(S∞(Tn)), Op(S−∞(Tn)), 338Am(Rn), Am

ρ,δ(Rn), 275

Am(Tn), Amρ,δ(T

n), A−∞(Tn), 341

Op(Am(Tn)), Op(Amρ,δ(T

n)), 341Op(A∞(Tn)), Op(A−∞(Tn)), 341S(Rn), 224Kers, 382sing supp, 383Hs(x), Hs(U), 383sing suppt, 383

Operators, etc.

clX(·), cl(·), U �→ U , 330‖ · ‖X , ‖ · ‖L(X,Y ), 330u �→ u, 302τx, R, 244(·, ·)L2(Tn) = (·, ·)H0(Tn), 302, 309‖ · ‖Hs(Tn), 307ϕs, 308(·, ·)Hs(Tn), 308〈·, ·〉, 308A∗, A(∗B), A(∗H), 309‖ · ‖s,t, 309�, �ξ, 310�, �ξ, 310‖ · ‖�p , 319δj,k, 322

∂kx , ∂

(k)x , ∂

(−k)x , 327

σ, A �→ σA, 335Op, σ �→ Op(σ), 335a �→ a, 342, 346a �→ a1, a �→ a2, 346f �→ f , FRn , 222F−1

Rn , 225FTn , F−1

Tn , 301exp, 368a∗, a(∗B), a(∗H), 370[·, ·], 371Lj , Rk, 423[·, ·]θ, 385σA, 262Dα

x , 327(−Dy)(α), 394

Notation 695

Other notation

p.v. 1x , 1

x±i0 , 238A|W , 330A(x)|x=x0 , 330uj

τ−→ u, 330Ind, 381dim, codim, 381z, 302〈ξ〉, 221, 300ξ(j), 313S

(j)k ,

{kj

}, 321

K, 340, 350

∼, m∼,m,ρ,δ∼ , 342

[·], 342σ ∼∑∞

j=0 σj , 352Op(σ) ∼∑∞

j=0 Op(σj), 352pj →∞, 386α!, 223α ≤ β, 223|α|, 223Dα, 224L, 225

Part III

K, 430Aut(V ), 431Aff(V ), 431xA, Ax, AB, A0, A−1, An, A−n, 432H < G, H � G, 432Z(G), 432GL(n, R), O(n), SO(n), 433GL(n, C), U(n), SU(n), 433G/H, 433H\G, 434Gq, 437U(H), 438πL, πR, 439φ ∼ ψ, 441τG/H , 447

HOM(G1, G2), 448⊕j∈J φ|Hj , 450

Haar(f), 459PG/H , 462HaarG/H , 463

G, 468TrigPol(G), 474L2(G), 477ResG

Hψ, 482Cφ(G,H), 484IndG

φH, 484exp(X), 492LieK(A), 499Lie(G), g, 499gl(n, K), o(n), so(n), u(n), su(n), 500SL(n, K), sl(n, K), 501Ad(A)X, 505ad(X)Y , 506, 508U(g), 507

Part IV

Re G, 530Hξ, ξ : G→ U(Hξ), 530f(ξ), 531φu, 531DY f , 532f(ξ)mn, 533δmn, 533LG, L, 534D′(G), 534〈·, ·〉G, 535Hs(G), 535Hξ, 536dim(ξ), 536λξ, 537M(G), 537L2(G), 537(·, ·)L2(G), 537〈ξ〉, λ[ξ], 538

696 Notation

S(G), pk, 539S ′(G), 543〈·, ·〉G, 543

Lp(G), �p(G, dimp( 2

p− 12 )

), 546

Lpk(G), 550

KA(x, y), LA(x, y), RA(x, y), 550l(f), r(f), 551, 579∂α, 534, 560σA(x, ξ), 552fφ, Aφ, 556uL, uR, 556〈A,B〉HS , ||A||HS , ||A||op, 559�α

ξ , 564�q, 564Am

k (M), 566Σm(G), Σm

k (G), 575πL, πR, 580RA(x), LA(x), 582la, la(x), rA, rA(x), 583D(G), 591SU(2), 599H, 603Sp(n), 606Sp(n, C), 606w1, w2, w3, 607Y1, Y2, Y3, 607D1, D2, D3, 609∂+, ∂−, ∂0, 611Vl, Tl, 612tl, tlmn, P l

mn, 617t−−, t−+, t+−, t++, 621f(l)mn, 632σA(x, l), σA(x, l)mn, 632σ∂+ , σ∂− , σ∂0 , 634Sm(SU(2)), 656Sm(S3), 661D′L1(M), 668pE→B , 668K\G, 669

Index

∗-algebra, 213Ψ(M), 423Diff(M), 419of pseudo-differential operators

on Tn, 380∗-homomorphism, 213〈ξ〉 on a group, 538⟨tl

⟩on SU(2), 633

μ-almost everywhere, 137μ-integrable, 143∂α on groups, 560σ-algebra, 119

Abel–Dini theorem, 386action

free, 668left, right, 437linear, 437of a group, 436transitive, 437, 669

adjoint operator, 107Banach, 101on a group, 569, 591

adjoints (Banach, Hilbert), 309Ado–Iwasawa theorem, 508affine group, 431algebra, 191∗-algebra, involutive, 213Spec(A), spectrum of, 208Banach, 200C∗-algebra, 213character of, 208commutative, 191derivations of, 499homomorphism, 195Hopf, 520

involution, 213isomorphism, 195Lie, 498quotient, 194, 199radical of, 193, 211semisimple, 193tensor product, 196topological, 196unit, inverse, 191unital, 191universal enveloping, 507

algebra of periodic ΨDOs, 367, 380algebra reformulation, 518

associativity diagram of, 518co-algebra, 519co-associativity diagram, 519multiplication mapping, 518tensor product, 518unit mapping, 518

algebraicbasis, 80dimension, 81number, 24tensor product, 84

almost orthogonality lemma, 95amplitude

of adjoints, 370of periodic integral operator, 388operator, 275, 341toroidal, 340

amplitudes Am(Rn), Amρ,δ(R

n), 275amplitudes Am(Tn), Am

ρ,δ(Tn), 340

Arzela–Ascoli theorem, 57asymptotic equivalence, 342

698 Index

asymptotic expansion, 352, 353of adjoint, 279, 370of parametrix, on a group, 577of parametrix, toroidal, 380of product, 371of transpose, 280, 369

asymptotic sums, 351atlas, 416automorphism, 430

inner, 531space Aut(V ), 431

automorphism group, 431Axiom of Choice, 18, 25, 73

for Cartesian products, 18

Baire’s theorem, 96balls Br(x), Bd(x, r), 26Banach

adjoint, 101algebra, 200duality, 308fixed point theorem, 43injective tensor product, 91projective tensor product, 91

Banach space, 94dual of, 101reflexive, 102

Banach–Alaoglu theorem, 99in Hilbert spaces, 109in topological vector spaces, 89

Banach–Steinhaus theorem, 97barrel, 90basis

algebraic, 80orthonormal, 110

Bernstein’s theorem, 306bijection, 13bilinear mapping, 83Borel

σ-algebra, 119measurable function, 135sets, 119

Borel–Cantelli lemma, 122boundary, 36

bounded inverse theorem, 99

C∗-algebra, 213Calderon–Zygmund covering lemma,

257canonical mapping of a Lie algebra,

507Caratheodory condition, 125Caratheodory–Hahn extension, 123cardinality, 20Cartan’s maximal torus theorem,

481Cartesian product, 12, 18, 71Casimir element, 510Cauchy’s inequality, 229Cauchy–Schwarz inequality, 103,

229, 230chain, 15character

of a representation, 479characterisation of S−∞(Tn), 348characteristic function, 13, 135characters, 582Chebyshev’s inequality, 148choice function, 17closed graph theorem, 99closure, 36closure operator, interior operator,

37co-algebra, 519

monoid, 521co-induced

family, 14topology, 69

commutant, 198commutator, 192, 371commutator characterisation

Euclidean, 414on a group, 566on closed manifolds, 421toroidal, 424

complemented subspace, 101complete topological vector space,

86

Index 699

completion, 44of a topological vector space, 86

component, 448composition, 13composition formula

Euclidean, 271for Fourier series operators, 394,

397on a group, 567, 568toroidal, 371

continuitymetric, 30topological, 46uniform on a group, 452

continuum hypothesis, 26generalised, 26

contraction, 43convergence

almost everywhere, 138almost uniform, 138in Lp(μ), 155in measure, 138in metric spaces, 29in topological spaces, 32metric uniform, 42of a net, 77pointwise, 41, 138uniform, 42, 138

convex hull, 89convolution, 228

left, right, l(f), r(f), 579associativity of, 228non-associativity of, 245of distributions, 244of linear operators, 520of sampling measures, 456on a group, 478, 532translations of, 246

convolution kernel, left, right, 582convolution operators, 551, 579Cotlar’s lemma, 406cover, 49

locally trivialising, 668cyclic vector, representation, 450

de Morgan’s rules, 12density, 36derivations of operator-valued

symbols, 587derivatives and differences, 325diameter, 28difference operators

forward, backward, 310on SU(2), �q, �+, �−, �0, 636on SU(2), formulae for, 638on a group, 564

Diracdelta, 239, 243delta comb, 306, 364

direct sum, 101, 106algebraic, 440of representations, 450

discretecone, 389fundamental theorem of calculus,

314integration, 314partial derivatives D

(α)x , 327

polynomials, 313Taylor expansion, 315

disjoint family, 119distance, 26

between sets, 28distribution function, 255distributionsD′(Ω), 242E ′(Ω), 242D′(Tn), periodic, 304on manifolds, 419periodic, 307, 308summable, D′L1(M), 668

dualalgebraic, 84Banach, Hilbert, 308of Lp(μ), 170of a Banach space, 101second, 102space, 85

700 Index

unitary, 468duality〈·, ·〉G, 535〈·, ·〉G, 543

Egorov’s theorem, 139ellipticity, 376

on a group, 577embedding, 309embedding theorem, 294endomorphism, 430equicontinuous family, 57, 85equivalence relation, 14Euler’s angles

on S3, 604on SO(3), 597on SU(2), 601

Euler’s identity, 284exponential coordinates, 500exponential of a matrix, 492extreme set, 89

family, 9family induced, co-induced, 14, 134Fatou’s lemma, 146

reverse, 147Fatou–Lebesgue theorem, 151fiber, 668fiber bundle, 668

principal, 668finite intersection property, 50, 72Fourier coefficients

on a group, 475Fourier coefficients, series, 302Fourier inversion

global, 580Fourier inversion formula

Euclidean, 225on S ′(Rn), 238on S(Zn), C∞(Tn), 301

Fourier serieson L2(Tn), 302on a group, 475

Fourier series operator, 393, 407

Fourier transformf(l)mn, on SU(2), 632and rotations, 227Euclidean, 222inverse, on S ′(G), 545inverse, on Lp(G), 548matrix, 533multiplication formula, 226of Gaussians, 226of tempered distributions, 233on D′(G), 545on L1(G), 548on Lp(G), 548on D′(Tn), 305on a group, 475on group G, 531toroidal, periodic, 301

Frechet space, 87Fredholm

integral equations, 44operator, 381

freezing principle, 288Frobenius reciprocity theorem, 488Fubini theorem, 187Fubini–Tonelli theorem, 186function, 12M-measurable, Borel measurable,

Lebesgue measurable, 135Holder continuous, 306harmonic, 293holomorphic, 217negative part of, 141periodic, 300positive part of, 141simple, 141test, 88weakly holomorphic, 204

functionalHaar, 454–460linear, 82positive, 175, 453positive, in C∗-algebra, 216

Index 701

functional calculus at the normalelement, 216

Gelfandtheorem, 1939, 203theorem, 1940, 208, 210theory, 207topology, 209, 517transform, 209, 517

Gelfand–Beurling spectral radiusformula, 205

Gelfand–Mazur theorem, 205Gelfand–Naimark theorem, 214

commutative, 215graph, 99group, 430

SU(2), 599SO(2), 596SO(3), 596Sp(n, C), 606Sp(n), 606U(1), 595unitary, U(H), 438action, 436affine, 431centre of, 432commutative, Abelian, 431, 491compact, 451finite, 431general linear GL(n, R),

GL(n, C), 433homomorphism, 435infinitesimal, 498isomorphism, 435Lie, 491linear Lie group, 491locally compact, 451orthogonal O(n), 433permutation, 432product, 431representation of, 439special linear SL(n, K), 501special orthogonal SO(n), 433special unitary SU(n), 433

symmetric, 431topological, 445transformation, right, left, 668unitary U(n), 433, 438

Holder’s inequality, 153, 230converse of, 173discrete, 319for Schatten classes, 113general, 231

Haarfunctional, 454–460integral, 454measure, 454

Haar integralon SO(3), 599on SU(2), 605

Hadamard’s principal value, 238Hahn decomposition, 161Hahn–Banach theorem, 96

in locally convex spaces, 88Hamel basis, 80Hausdorff

maximal principle, 18, 73space, 53total boundedness theorem, 86

Hausdorff–Young inequality, 236,304on G and G, 548

Heaviside function, 239Heine–Borel property, 90Heine–Borel theorem, 59Hilbert

duality, 308space, 103

Hilbert–Schmidt, 559, 662operators, 112spectral theorem, 109

homeomorphism, 48homomorphism, 195, 430

continuous, 448differential, 502space HOM(G1, G2), 448

702 Index

Hopf algebra, 520“everyone with the antipode”

diagram, 521and C∗-algebra, 524antipode, 520co-multiplication and unit

diagram, 521for compact group, 523for finite group, 522multiplication and

co-multiplication diagram, 521multiplication and co-unit

diagram, 521tensor product, 520

Hopf fibration, 673hyperbolic equations, 410hypoellipticity, 384, 385

idealspanned by a set, 193two-sided, maximal, proper, 193

index, 388of Fredholm operator, 381

index sets, 11induced

family, 14induced representation space

IndGφH, 484

injection, 13inner product, 103

on V ⊗W , 84integral, 143

Haar, 454Lebesgue, 143Pettis, weak, 90, 482Riemann, 151

integrationdiscrete, 314

interior, 36interpolation theorems, 385invariant

vector fields, 500isometry, 443

isomorphism, 195, 430canonical, 308intertwining, 531isometric, 48

isotropy subgroup Gq, 437

Jacobi identity, 361, 498for Gaussians, 361

Jordan decomposition, 159, 160

kernelof a linear operator, 82

kernel, null space, 430, 435kernels la, la(x), rA, rA(x), 583Killing form, 509Krein–Milman theorem, 89Krull’s theorem, 193Kuratowski’s closure axioms, 37

Laplace operator, 225on SU(2), 611, 625on a group, 512, 534on a group, symbol of, 554

law of trichotomy, 21Lebesgue

Lp(μ)-norm, 152Lp(μ)-spaces, 154–B.Levi monotone convergence

theorem, 144conjugate, 153covering lemma, 61decomposition of measures, 168differentiation theorem, 252, 253dominated convergence theorem,

150, 222integral, 143measurable function, 135measure, 128measure, translation and rotation

invariance, 129measurelet, 117non-measurable sets, 133outer measure, 117space Lp(μG) on a group, 460

left quotient H\G, 434

Index 703

Leibniz formulaasymptotic, 571discrete, 311Euclidean, 249on an algebra, 499

LF-space, 88Lie

algebra, 498algebra sl(n, K), 501algebra homomorphism, 498algebra of a Lie group, 499algebra, canonical mapping, 507algebra, semisimple, 509algebras gl(n, K), o(n), so(n),

u(n), su(n), 500group, 491group, dimension of, 499group, exponential coordinates,

500group, linear, 491group, semisimple, 509subalgebra, 498

lifting of operators, 673linear operator

bounded, 94compact, 95norm of, 94

Liouville’s theoremfor harmonic functions, 293for holomorphic functions, 217

Littlewood’s principles, 142locality, 265, 382logarithm of a matrix, 495Luzin’s theorem, 141

manifold, 65, 417closed, 419differentiable, 66orientable, 418paracompact, 423

mapping, 12continuous, uniformly continuous,

Lipschitz continuous, 49measurable, 134

Marcinkiewicz’ interpolationtheorem, 256

maximum, minimum, supremum,infimum, 15

measure, 121absolutely continuous, 163action-invariant on G/H, 463Caratheodory–Hahn extension of,

123Haar, 454Hahn decomposition of, 161Jordan decomposition of, 159Lebesgue, 128Lebesgue decomposition of, 168outer, 116probability, 122product of, 181Radon–Nikodym derivative of,

162sampling, 455semifinite, 158signed, 158variations of, positive, negative,

total, 159measure space, 122

σ-finite, 164Borel, 122complete, 122finite, 122

measurelet, 116measures

mutually singular, 161metric, 26

discrete, 27Euclidean, 27interior, closure, boundary, 35subspace, 27sup-metric, d∞, 27

metric spacecomplete, 40sequentially compact, 58totally bounded, 61

metricsequivalent, 49

704 Index

Lipschitz equivalent, 33Minkowski’s functional, 87Minkowski’s inequality, 153, 231

for integrals, 188mollifier, 251monoid, 456Montel space, 90multi-indices, 223multiplication of distributions, 238

Napier’s constant, 40neighbourhood, 29, 36net, 77

Cauchy, 86neutral element, inverse, 431norm, 92

equivalent, 93operator, 94trace, 111

normaldivisors, 432element, 216, 517subgroup, 432

nuclear space, 92numbers, 10

algebraic, 24Stirling, 321

open mapping, 98open mapping theorem, 98operator

Dα, 224Dα

x , D(α)x , 327

(−Dy)(α), 394adjoint, 107classical, 355compact, 193intertwining, 441left-invariant, right-invariant, 551linear, 82order of, 414, 420properly supported, 280self-adjoint, 107

operator norm, 94

operatorsHilbert–Schmidt, 112trace class, 111

operators ∂+, ∂−, ∂0, 611applied to tl, 628symbols of, 634in Euler angles, 611

orderpartial, 15total, linear, 15

orthogonal projection, 105outer measure, 116

Borel regular, 124metric, 125product of, 181

parallelogram law, 105parametrix, 287, 378, 380

on a group, 577Parseval’s identity

on Rn, 236on a group, 475, 476, 538

partition of unity, 56path, 67Pauli matrices, 607Peetre’s

inequalities, 321theorem, 266

periodicSchwartz kernel, 336Taylor expansion, 328

periodic integral operator, 387periodicity, 300periodisation, 360

compactly supportedperturbations, 366

of operators, 363of symbols, 365

Peter–Weyl theorem, 470for Tn, 471left, 471

Pettis integral, 90, 482Plancherel’s identity, 302

in Hilbert space, 110

Index 705

on Rn, 236on a group, 475, 476

pointaccumulation, 36, 52fixed, 43isolated, 36

Poisson summation formula, 361polynomial

discrete, 313trigonometric, TrigPol(G), 474trigonometric, on Tn, 305

Pontryagin duality, 469power set P(X), 10preimage, 30preimage, image, 13principal symbol, 352principle

convergence, 234Littlewood, 142uniqueness, 234

productof measures, outer measures, 181topology, 34, 40, 72

product group, 431projection PG/H , 462pseudo-differential operator

local, 414on a manifold, 418periodic, continuity of, 343toroidal, 338

pseudo-differential operatorsΨm(G), 573Ψm(M), 418Ψm(SU(2)), Ψm(SU(2)), 528on a group, 553

pseudolocality, 265, 383pushforwards, φ–, 556Pythagoras’ theorem, 103

quantizationon Rn, 276on a group, 552operator-valued, 583toroidal, 336

quantum numbers, 630, 661quaternions, 603, 660quotient

algebra, 194, 199left H\G, 434right G/H, 433topology, 69, 199topology on G/H, 447vector space, 82

Radon–Nikodym derivative,theorem, 162, 164

relation, 12Rellich’s theorem, 295representation

tl on SU(2), 617regular, left, right, 551, 580adjoint, of a Lie algebra, 506adjoint, of a Lie group, 505cyclic, 450decomposition of, 468dimension dim (ξ), 530dimension of, 439direct sum of, 450equivalent, 441induced, 484irreducible, 440matrix, 469multiplicity of, 488of a group, 439regular, left, right, πL, πR, 439,

470restricted, 440space IndG

φH, 484space Rep(G), 530strongly continuous, 449topologically irreducible, 450unitary, 439unitary matrix, 439

resolvent mapping, 204restriction, 13Riemann integral, sums, 151Riemann–Lebesgue lemma, 223

706 Index

Rieszalmost orthogonality lemma, 95representation theorem, 107topological representation

theorem, 175, 177Riesz–Thorin interpolation theorem,

158right quotient G/H, 433right transformation group, 668Russell’s paradox, 11

scaling, 241Schatten class, 113Schroder–Bernstein theorem, 21Schrodinger equation, 412Schur’s lemma, 284, 443Schwartz kernel, 92, 340, 350, 550,

592periodic, 336

Schwartz kernel theorem, 92Schwartz space, 87, 224S(Rn), 224S(Zn), 300

Schwartz’ impossibility result, 238semigroup, 456seminorm, 92separability, topological, 36separating points, 55, 75sequence, 29

Cauchy, 40generalised, 77

sequential density of functions, 240set

directed, 77sets, 9, 10

ψ-measurable, 118balanced, 80Borel, 119bounded, 28compact, 49convex, 80elementary, 116extreme, 89Lebesgue non-measurable, 133

linearly independent, 80open, 29open, closed, 31well-ordered, 16

singular support, 243, 245, 383small sets property, 85small subgroups, 496smooth mapping, 417smoothing operators, 266smoothing periodic ΨDOs, 347Sobolev spaces, 293

Hs(G), 535, 542Lp

k(G), 550Lp

k(Ω), 247Lp

s(Tn), 375

biperiodic, 309localisation, Lp

k(Ω)loc, 248on manifolds, Hs(M), 419toroidal, Hs(Tn), 307

spaceC∞(Tn × Zn), 338Ck(M), C∞(M), 417C∞0 (Ω), 239L2(Tn), 302L2(G), 477Lp

loc(Rn), 241

Lp(G), 546Lp(Rn), 221Lp(Rn), interpolation, 232Lp(Tn), 304Pol1(SU(2)), 657K-vector, 79Ψm(M), 418D′(G), 534, 591D′(M), 419D(G), 591D(M), 419S ′(G), 543M(G), 537S(G), 539Diff(M), 419barreled, 90base, 668

Index 707

homogeneous, 669measure, 122metric, 26quotient, 14simply-connected, 504topological, 31total, 668

span, 80spectral radius formula, 205spectrum

of an algebra element, σ(x), 192of an algebra, Spec(A), 517of an operator, σ(A), 82

Stein–Weiss interpolation, 549Stirling numbers, 321

recursion formulae, 322Stone–Weierstrass theorem, 63subalgebra

involutive, 63subcover, 49subgroup, 432

trivial, 432subnet, 77subspace

compact, 50invariant, 440metric, 27, 34trivial, 80, 440vector, 80

subspacesorthogonal, 103

sum, infinite sum, 115summation by parts, 313summation on SU(2), 630support, 55, 243surjection, 13Sweedler’s example, 526symbol

classical, 283, 355elliptic, 289Euclidean, σA, 262homogeneous, 283of periodic ΨDO, 357on a group, 552

operator-valued, left, right, 583principal, 352toroidal, 335, 336

symbol classSm

ρ,δ(Rn × Rn), 261

Smρ,δ(T

n × Rn), 338Sm

ρ,δ(Tn × Zn), 338

Sm(S3), 661Sm(SU(2)), 656Σm(G), 575Σm(SU(2)), 633Σm

0 (SU(2)), Σmk (SU(2)), 633

Taylor expansionbiperiodic, 328discrete, 315on a group, 561periodic, 327, 328

tempered distributionsS ′(Rn), 233S ′(Zn), 300

tensor productalgebra, 196, 518algebraic, 84Banach, injective, 91Banach, projective, 91Hopf algebra, 520injective, 91of operators, 84of spaces, 83projective, 90spaces, dual of, 84

test functions, 88, 300Tietze’s extension theorem, 71Tihonov’s theorem, 73topological

algebra, 196equivalence, 48group, 445interior, closure, boundary, 36property, 48vector space, 85zero divisor, 203

708 Index

topological approximationof Lebesgue measurable sets, 131of measurable sets, 126

topological spacecompact, 49complete, 86completion of, 86connected, disconnected, 66Hausdorff, 53locally compact, 49locally convex, 87paracompact, 423path-connected, 67separable, 36totally bounded, 86

topology, 31F-induced, 71base of, 39co-induced, 69discrete, 75induced, 48injective tensor, 91metric, 36, 39metric, canonical, 32metric, comparison, 32metrisable, 74norm, 94on R2, 34product, 34, 40, 71projective tensor product,

π-topology, 90quotient, 69, 199quotient on G/H, 447relative, 33second countable, 39strong operator, 449subbase, subbasis, 39weak, 88, 109weak∗, 88, 99

toroidalamplitude, 340

torus, 299inflated, 362

tower, 19

trace, trace class, trace norm, 111transfinite induction, 17

mathematical induction, 17transpose, 279transposed operator

on a group, 570, 591triangle inequality, 26, 104trigonometric polynomials

TrigPol(G), 474TrigPol(Tn), 305

uncertainty principle, 241uniform boundedness principle, 97unit, 191unital algebra, 191unitary dual G, 468, 530universal enveloping algebra, 507

as Hopf algebra, 526universality

of enveloping algebra, 507of permutation groups, 436of unitary groups, 491

Urysohn’s lemma, 55smooth, 254

vector space, 79Banach, 94dimension of, 81Frechet, 87Hilbert, 103inner-product, 103LF-, 88locally convex, 87Montel, 90normed, 92nuclear, 92quotient, 82topological, 85

vectorslinearly independent, 80orthogonal, orthonormal, 103

Vitali’s convergence theorem, 156

Index 709

wave front set, 390weak derivative, 246weak topology, 88, 109weak type (p, p), 256weak∗-topology, 88, 99Weierstrass theorem, 62Well-ordering principle, 17, 25Whitney’s embedding theorem, 418

Young’s inequality, 187discrete, 320for convolutions, 231

general, 232on Rn, 229

Zermelo–Fraenkel axioms, 26Zorn’s lemma, 19