- ↑—monotonically converges from below, 14.2
- є/3 argument, 8.1
- B(X), 6.3
- B(X,Y), 6.3
- CP[−π,π], 5.1
- c0, 11.1
- F(X,Y), 8.1
- H2, A.4
- K(X,Y), 8.1
- L1, 13.2
- L2[a,b], 2.3
- L∞, 13.2
- L, 12.2
- L(X), 6.3
- L(X,Y), 6.3
- Lp, 14.1
- l2, 2.2, 11.1
- l∞, 11.1
- lp, 11.1
- S, Schwartz space, 15.4
- S(X), 13.2
- IX, 6.1
- kerT, 6.1
- 1X (the identity map on X), 16.2.3
- supp, 14.4
- ||·||1 norm, 1.1.1, 1.1.1, 2.1
- ||·||2 norm, 1.1.1, 1.1.1, 2.1
- ||·||∞ norm, 1.1.1, 1.1.1, 2.1
- ⊥, 3.1
- σ-additivity, see countable additivity
- σ-algebra, 12.1
- σ-finite
- σ-ring, 12.1
- ⊔, 12.1
- d1 metric, 1.1.1, 1.1.1, 1.1.1
- d2 metric, 1.1.1, 1.1.1
- d∞ metric, 1.1.1, 1.1.1
- S(k), 15.1
- Z, 6.1
- CLin(A), 2.4
- l1n, 2.1
- l2n, 2.1
- l∞n, 2.1
- Cb(X), 2.1
- l∞(X), 2.1
- Lin(A), 2.4
- “zigzag” function, 16.3.1
- a.e., see almost everywhere
- absolute continuity, 13.3
- absolutely continuous charge, 13.5
- abstract completion of metric space, 1.2.1
- accumulation point, 1.2.2
- additivity, 12.1
- adjoint
operator, 6.4
- adjoints, C.2
- algebra
- almost everywhere, 12.2
- alternative
- analysis, 2.3
- analytic function, D.1
- approximation, 3.2
- by polynomials, 5.4
- identity, of the, 5.2, 15.4
- Weierstrass, of, 5.4
- argument
- average
- axiom of choice, 11.4
- Baire’s categories, 16.3.1
- Baire’s category theorem, 16.3.1
- Banach Fixed point theorem, 16.2.1
- Banach space, 2.1, 11.1, C.1
- Banach–Steinhaus Uniform Boundedness theorem, 16.3.2
- Bessel’s inequality, 3.2, C.1
- Borel σ-algebra, 14.3
- Borel set, 14.3
- ball
- basis, C.2
- best approximation, C.1
- bilateral right shift, A.4
- bounded
- bounded linear
functional, 4.1
- bounded linear
operator, 6.1
- Cantor
- Cantor function, 2.3
- Carathéodory
- Cauchy integral formula, 5.3
- Cauchy sequence, 1.2.1, 2.1
- Cauchy–Schwarz inequality, 1.1.1, C.1
- Cauchy–Schwarz–Bunyakovskii inequality, 2.2
- Cayley
transform, 7.3
- Cayley transform, C.2
- Cesàro sum, 5.1
- Chebyshev
- Chebyshev polynomials, 3.4
- Closed Graph theorem, 16.3.4
- calculus
- category
- category
theory, 2
- character, 15.2
- charge, 12.3
- charges
- closable
- closed
- closed linear span, 2.4
- closure, 1.1.2
- coefficient
- coefficients
- coherent states, 5.3
- compact, 1.2.2
- compact operator, 8.1, C.2
- singular value decomposition, 9.2
- compact set, 8.1
- complement
- complete
- complete metric
space, 2.1
- complete o.n.s., C.1
- complete orthonormal sequence, 3.3
- complete space, 1.2.1
- condition
- conditions
- continuity
- continuous
- continuous on average, 14.2
- contraction, 16.2.1
- convergence
- almost everywhere, 13.1
- in measure, 13.1
- monotone
- theorem B. Levi, on, 13.3
- uniform, 13.1
- convergent
- convex, 3.1
- convex set, 2.1, 16.2.1
- convolution, 15.1
- convolution operator, 15.1
- coordinates, 2
- corollary about orthoprojection, 6.2
- cosine
- Fourier coefficients, 0.1.2
- countable
- countably
- countable sub-additivity, 12.2
- countably additive
- cover
- Dirichlet kernel, 16.3.2
- decreasing
- dense
- derivative, D.1
- diagonal argument, 8.1
- diagonal operator, 6.5
- diffeomorphism, 16.2.3
- differentiable
function, D.1
- differential equation
- separation of variables, 0.2.1
- discrete
- disjoint
- disjunctive measures, 12.3
- distance, see metric, 2, 2.1
- distance function, 2.1
- domain
- dual group, 15.2
- dual space, 4.1
- dual spaces, C.2
- duality
- Egorov’s theorem, 13.1
- eigenspace, 9.1
- eigenvalue of operator, 7.1
- eigenvalues, C.2
- eigenvector, 7.1
- equation
- Fredholm, 10
- first kind, 10
- second kind, 10, 10
- heat, 5.4
- Volterra, 10
- equivalent
- equivalent charges, 13.5
- essentially bounded function, 13.2
- examples of Banach spaces, C.1
- examples of Hilbert spaces, C.1
- Fatou’s lemma, 13.3
- Fejér
- Fejér kernel, 5.2, C.1
- Fejér sum, 5.1
- Fejér’s theorem, C.1
- Fourier
- Fourier
transform
- Fourier
analysis, 0.3
- Fourier coefficient, 3.3
- Fourier coefficients, 0.1.2
- Fourier series, 0.1.2, C.1
- Fourier, Joseph, 0.3
- Fredholm
equation, 10
- Fredholm alternative, 10, C.2
- Fredholm equation
- Fredholm equation of the second
kind, 10
- Fubini theorem, 13.4
- finite
- finite rank operator, 8.1, C.2
- first category, 16.3.1
- first resolvent
identity, 7.1
- fixed point, 16.2.1
- formula
- integral
- Parseval’s, of, 5.3
- frame of references, 2
- function
- “zigzag” , 16.3.1
- analytic, D.1
- bounded
- Cantor, 2.3, 12.4
- differentiable, D.1
- essentially bounded, 13.2
- generating, 15.3
- holomorphic, D.1
- indicator, 13.2
- integrable, 13.2
- seesummable function, 13.2
- measurable, 13.1
- rapidly decreasing, 15.4
- simple, 13.2
- square integrable, 2.3
- step, 14.2, 15.5
- summable, 13.2, 13.2
- support, 14.4
- functional, see linear functional
- functional calculus, 7
- functions of operators, 7
- fundamental domain, 0.1.1
- Gaussian, 15.4, 15.5
- Gram–Schmidt orthogonalisation, 3.4
- Gram–Schmidt orthonormalization process, C.1
- general compact operators, C.2
- generating function, 15.3
- graph
- group
- group representations, 5.3
- Hölder’s Inequality, 11.1
- Haar measure, 15.1
- Hahn decomposition of a charge, 12.3
- Hahn-Banach theorem, 11.4
- Hardy space, A.4
- Heine–Borel
theorem, 8.1
- Heine–Borel theorem, 1.2.2
- Hermitian
operator, 6.5
- Hermitian operator, C.2
- Hilbert
space, 2.2
- Hilbert space, C.1
- Hilbert–Schmidt
operator, 8.2
- Hilbert–Schmidt norm, 8.2, A.6
- Hilbert–Schmidt operators, C.2
- Hilbert–Schmidt theory, C.2
- heat equation, 5.4
- holomorphic
function, D.1
- Inverse, C.2
- Inverse Function theorem, 16.2.3
- identity
- identity
operator, 6.1
- implicit function theorem, 16.2.3
- incomplete spaces, C.1
- indicator function, 13.2
- inequality
- Bessel’s, 3.2
- Cauchy–Schwarz, 1.1.1
- Cauchy–Schwarz–Bunyakovskii, of, 2.2
- Chebyshev, 13.3
- Hölder’s, 11.1
- Minkowski’s , 11.1
- triangle, of, 2.1, 2.1
- inner product, 1.1.1, 2.2
- inner product space, 2.2
- complete, see Hilbert space
- inner-product space, C.1
- integrability conditions, ??
- integrable
- function, 13.2
- seesummable function, 13.2
- integral
- integral
formula
- integral equations, C.2
- integral operator, 8.2, 10
- with separable kernel, 10
- interior, 1.1.2
- invariant measure, 15.1
- inventor’s paradox, 16.1
- inverse
operator, 6.3
- inverse Fourier transform, 15.5
- inverse image, 1.1.3
- invertible operator, 6.3
- isometric
- isometric metric space, 1.1.1
- isometry, 1.1.1, 6.5, 11.2
- isomorphic
- isomorphic spaces, 11.2
- isomorphism, 11.2, A.5
- Jacobian, 16.2.1
- kernel, 10
- kernel of convolution, 15.1
- kernel of integral
operator, 8.2
- kernel of linear functional, 4.1
- kernel of linear operator, 6.1
- Laguerre polynomials, 3.4
- Lebesgue
- integral, 13.2
- measure
- set
- theorem, 12.2
- theorem on dominated convergence, 13.3
- Lebesgue
integration, 2.3
- Lebesgue measure, 12.4
- Legendre polynomials, 3.4
| - Levi’s theorem on monotone convergence, 13.3
- Lipschitz condition, 16.2.2
- ladder
- Cantor, see Cantor function
- leading particular case, 16.1, 16.2.3
- left
inverse, 6.3
- left shift operator, 6.3
- lemma
- about inner product limit, 2.4
- Fatou’s, 13.3
- Riesz–Fréchet, 4.2
- Urysohn’s , 14.3
- Zorn, 11.4
- length of a vector, 2
- limit
- linear
- linear
functional
- linear
operator
- linear
space, 2
- linear functional, 4.1, C.2
- linear operator, C.2
- linear span, 2.4
- local-Ck- diffeomorphism, 16.2.3
- locally compact topology, 15.1
- locally convex topological vector space, 16.4
- locally invertible function, 16.2.3
- Minkowski’s inequality, 11.1
- map
- mathematical way
of thinking, 3
- mathematical way of
thinking, 2
- mean value theorem, 16.2.1
- measurable
- measure, 12.1
- σ-finite, 12.1, 12.2
- absolutely continuous, 13.5
- complete, 12.2
- disjunctive, 12.3
- finite, 12.1
- Haar, 15.1
- invariant, 15.1
- Lebesgue, 12.4
- outer, 12.2
- product, 12.4, 13.4
- regular, 14.3
- signed, see charge
- metric, 1.1.1, 2.1, 11.1
- metric
space, 2
- metric space, 1.1.1
- monotonicity
- monotonicity of integral, 13.2
- multiplication operator, 6.1
- Neumann series, 7.1, 10, C.2
- nearest point theorem, 3.1
- neighbourhood, 1.1.2
- nilpotent, A.5
- norm, 1.1.1, 2.1, 11.1, C.1
- seesup-norm, 11.1
- ||·||1, 1.1.1, 1.1.1, 2.1
- ||·||2, 1.1.1, 1.1.1, 2.1
- ||·||∞, 1.1.1, 1.1.1, 2.1
- equivalent, 16.3.3
- Hilbert–Schmidt, 8.2, A.6
- sup, 11.1
- norm of linear operator, 6.1
- normal operator, 6.5, C.2
- normed
space, 2.1
- normed space, 1.1.1
- complete, see Banach space
- nowhere dense set, 16.3.1
- open
- open mapping theorem, 16.3.3
- operator, 11.2
- adjoint, 6.4
- bounded, 11.2
- closable, 16.3.4
- closed, 16.3.4
- closure, 16.3.4
- compact, 8.1
- singular value decomposition, 9.2
- convolution, 15.1
- diagonal, 6.5
- domain, 16.3.4
- eigenvalue of, 7.1
- eigenvector of, 7.1
- finite rank, 8.1
- graph, 16.3.4
- Hermitian, 6.5
- Hilbert–Schmidt, 8.2
- identity, 6.1
- integral, 8.2, 10
- kernel of, 8.2
- with separable kernel, 10
- inverse, 6.3
- invertible, 6.3
- isometry, 6.5
- linear, 6.1
- nilpotent, A.5
- normal, 6.5
- of
multiplication, 6.1
- self-adjoint, see Hermitian operator
- shift
- shift on a group, 15.1
- spectrum of, 7.1
- unitary, 6.5
- Volterra, A.4
- zero, 6.1
- operators
- orthogonal
- complement, 3.5
- projection, 6.2
- orthogonal
complement, 3.5
- orthogonal
polynomials, 3.4
- orthogonal
projection, 6.2
- orthogonal complements, C.1
- orthogonal sequence, 3.1, C.1
- orthogonal system, 3.1
- orthogonalisation
- orthogonality, 2.2, 3, C.1
- orthonormal
basis, 3.3
- orthonormal basis
- orthonormal basis
(o.n.b.), C.1
- orthonormal sequence, 3.1
- orthonormal sequence (o.n.s.), C.1
- orthonormal system, 3.1
- orthoprojection, 6.2
- outer measure, 12.2
- Parseval’s
- Parseval’s formula, C.1
- Picard iteration, 16.2.2
- Picard–Lindelöf theorem, 16.2.2
- Plancherel
- Pontryagin’s duality, 15.2
- Pythagoras’ school, 5.4
- Pythagoras’ theorem, 3.1
- Pythagoras’s theorem, C.1
- pairwise
- parallelogram identity, 2.2, C.1
- partial sum of the Fourier series, 5.1
- period, 0.1.1
- periodic, 0.1.1
- perpendicular
- point
- polynomial
- polynomial approximation, 5.4
- polynomials
- positive
- pre-image, 1.1.3
- product
- product measure, 12.4, 13.4
- projection
- quantum mechanics, 2, 2.3
- Radon–Nikodym theorem, 13.5
- Riesz representation, 14.4
- Riesz–Fischer theorem, C.1
- Riesz–Fisher theorem, 3.3
- Riesz–Fréchet lemma, 4.2
- Riesz–Fréchet theorem, C.2
- radius
- regular charge, 14.3
- regular measure, 14.3
- representation
- resolvent, 7, 7.1
- resolvent set, 7.1
- right
shift operator, 6.1
- right inverse, 6.3
- Schwartz space, 15.4
- Segal–Bargmann
space, 2.3
- Stone’s theorem, 16.1
- Stone–Weierstrass theorem, 16.1
- scalar product, 2.2
- school
- second category, 16.3.1
- self-adjoint operator, see Hermitian operator, C.2
- semi-norm, 16.4
- semiring, 12.1
- separable Hilbert space, 3.4
- separable kernel, 10, C.2
- separable metric space, 16.1
- separation of variables, 0.2.1
- sequence
- sequential continuity, 1.1.3
- sequentially compact, 1.2.2
- series
- set
- compact, 8.1
- Borel, 14.3
- bounded, 1.2.2
- Cantor, 12.2, 12.4
- closed, 1.1.2
- convex, 2.1, 3.1, 16.2.1
- dense, 1.1.2
- measurable
- nowhere dense, 16.3.1
- open, 1.1.2
- resolvent, 7.1
- symmetric difference, 12.2
- shift
- shift operator, 15.1
- signed measure, see charge
- simple function, 13.2
- sine
- Fourier coefficients, 0.1.2
- singular value decomposition of compact operator, 9.2
- space
- Banach, 2.1, 11.1
- complete, 1.2.1
- dual, 4.1
- Hardy, A.4
- Hilbert, 2.2
- inner product, 1.1.1, 2.2
- complete, see Hilbert space
- linear, 2
- locally convex, 16.4
- metric, 1.1.1, 2
- normed, 1.1.1, 2.1
- complete, see Banach space
- of bounded linear operators, 6.3
- Schwartz, 15.4
- Segal–Bargmann, 2.3
- separable metric, 16.1
- vector, see linear space
- space of finite sequences, 2.3
- span
- spectral
radius:, C.2
- spectral properties of normal operators, C.2
- spectral radius, 7.2
- spectral radius formula, C.2
- spectral theorem for compact normal operators, 9.2, C.2
- spectrum, 7.1, C.2
- statement
- Fejér, see theorem
- Gram–Schmidt, see theorem
- Riesz–Fisher, see theorem
- Riesz–Fréchet, see lemma
- step
- sub-additive
- countable sub-additivity, 12.2
- subcover, 1.2.2
- subsequence
- convergent
- quickly convergent, 13.2, 13.3
- subspace, 2.3
- subspaces, C.1
- sum
- Cesàro, of, 5.1
- Fejér, of, 5.1
- summable
- sup-norm, 11.1
- support of function, 14.4
- symmetric difference of sets, 12.2
- synthesis, 2.3
- system
- orthogonal, 3.1
- orthonormal, 3.1
- theorem
- Baire’s category, 16.3.1
- Banach fixed point, 16.2.1
- Banach–Steinhaus Uniform Boundedness, 16.3.2
- closed graph, 16.3.4
- Egorov, 13.1
- Fejér, of, 5.2
- Fubini, 13.4
- Gram–Schmidt, of, 3.4
- Hahn-Banach, 11.4
- Heine–Borel, 1.2.2, 8.1
- implicit function, 16.2.3
- inverse function, 16.2.3
- Lebesgue, 12.2
- Lebesgue on dominated convergence, 13.3
- mean value, 16.2.1
- monotone convergence, B. Levi, 13.3
- on nearest point , 3.1
- on orthonormal basis, 3.3
- on perpendicular, 3.2
- open mapping, 16.3.3
- Picard–Lindelöf , 16.2.2
- Pythagoras’, 3.1
- Radon–Nikodym, 13.5
- Riesz–Fisher, of, 3.3
- Stone’s, 16.1
- Stone–Weierstrass, 16.1
- spectral for compact normal operators, 9.2
- Weierstrass approximation, 5.4, 16.1
- Zermelo, 11.4
- thinking
- topology
- transform
- triangle
inequality, 2.1
- triangle inequality, 2.1, C.1
- trigonometric
- two monotonic limits, 13.1, 13.3, 13.4
- Urysohn’s lemma, 14.3
- uniform convergence, 13.1
- uniformly
- unit ball, 2.1
- unitary
operator, 6.5
- unitary operator, C.2
- Volterra equation, 10
- Volterra operator, A.4
- variation of a charge, 12.3, 14.3
- vector
- vector space, 2
- vectors
- Weierstrass approximation theorem, 5.4, 16.1, C.1
- wavelet
transform, 5.3
- wavelets, 5.3, 5.4
- windowed
Fourier transform, 5.4
- Zermelo’s theorem, 11.4
- Zorn’s Lemma, 11.4
- zero
operator, 6.1
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