Index of terminology

Absorbent set, absorption of one set by another : I, p. 7. Adapted bornology : III, p. 3. Adjoint : V, p. 38. Affine transformation : IV, p. 39. Associated (Hausdorff vector space) with a topological vector space : I, p. 4.

Balanced convex closed envelope of a set : II, p. 13 and p. 62. Balanced core of a set : I, p. 7. Balanced envelope of a set : I, p. 7. Balanced set : I, p. 6. Banach-Dieudonné theorem : IV, p. 24. Banach-Saks-Kakutani theorem : V, p. 68, exerc. 33. Banach space : I, p. 5. Banach-Steinhaus theorem : III, p. 25. Banach’s theorem : I, p. 17. Barrel : III, p. 24. Barrelled space : III, p. 24. Base of a bornology : III, p. 1. Basis (algebraic) of a Hilbert space : V, p. 22. Basis (Banach) : IV, p. 69, exerc. 14. Basis (complete Banach, contracting Banach) : IV, p. 70, exerc. 15. Basis (orthonormal) : V, p. 22. Basis (unconditional Banach) : IV, p. 71, exerc. 16. Bessel’s inequality : V, p. 21. Bidual : IV, p. 14. Bipolar theorem : II, p. 44. Birkhoff-Alaoglu theorem : V, p. 78, exerc. 13. Bishop-Phelps theorem : II, p. 77, exerc. 4. Bornological locally convex space : III, p. 12. Bornology : III, p. 1. Bornology (adapted) : III, p. 3. Bornology (canonical) : III, p. 3. Bornology (convex) : III, p. 2. Bornology generated by a family of sets : III, p. 1. Bornology (product) : III, p. 2. Bornivorous set : III, p. 39, exerc. 11. Bounded set : III, p. 2 and p. 37, exerc. 1.

Canonical bornology : III, p. 3. Canonical mapping of $\bigoplus_{i \in I} E_i'$ in $\left( \prod_{i \in I} E_i \right)'$ : IV, p. 13. Canonical mapping of E in $E''$ : IV, p. 14. Canonical mapping of E onto $E'$ (E a Hilbert space) : V, p. 15. Canonical topology on a finite dimensional vector space : I, p. 2. Cap : II, p. 57. Cauchy-Schwarz inequality : V, p. 3. Closed graph theorem : I, p. 19. Closed half-spaces defined by a closed hyperplane : II, p. 15. Cobord : IV, p. 72, exerc. 3. 1-cocycle (continuous) : IV, p. 72, exerc. 3.

Compact linear mapping : III, p. 6. Compatible topology and structure of an ordered vector space : II, p. 15. Compatible vector space structure and preorder : II, p. 12. Compatible vector space structure and topology : III, p. 1. Compatible with the duality (locally convex topology) : IV, p. 1. Complement (orthogonal) : V, p. 13. Complete topological vector space : I, p. 5. Completion of a Hausdorff prehilbertian space : V, p. 8. Completion of a Hausdorff topological vector space : I, p. 6. Complex linear form : II, p. 61. Complex linear variety : II, p. 61. Complex locally convex space : II, p. 62. Complexification (prehilbertian space) : V, p. 5. Complexified topological vector space : II, p. 62. Concave function : II, p. 17. Cone (asymptotic) : II, p. 67. Cone (convex) generated by a set : II, p. 11. Cone (pointed and non-pointed) : II, p. 10. Cone (polyhedral) : II, p. 91. Cone (proper pointed convex) : II, p. 11. Conjugate of a complex prehilbertian space : V, p. 6. Convex balanced envelope of a set : II, p. 10 and p. 62. Convex bornology : III, p. 2. Convex closed envelope of a set : II, p. 13. Convex function : II, p. 17. Convex set : II, p. 7 and p. 62. Convex (symmetric) envelope of a set : II, p. 16. Coordinates with respect to an orthonormal base : V, p. 22. Core (balanced) of a set : I, p. 7.

Density of order : V, p. 7. Dimension (hilbertian) : V, p. 24. Dimension of a convex set : II, p. 10. Dirichlet space : V, p. 8. Distal set : IV, p. 72, exerc. 1. Distinguished space : IV, p. 52, exerc. 4. Dual (algebraic) of a real topological vector space : II, p. 42. Dual of a locally convex space (real or complex) : III, p. 14. Dual of a real topological vector space : II, p. 42. Dual (weak, strong) : III, p. 14. Duality separating in F, separating duality : II, p. 41. Duality (vector spaces in) : II, p. 40. Dvoretzky-Rogers theorem : V, p. 63, exerc. 14.

Eberlein’s theorem : IV, p. 35. D. Edwards’ theorem : II, p. 94, exerc. 41. Endomorphism (hermitian) : V, p. 44. Endomorphism (normal) : V, p. 43. Endomorphism (positive) : V, p. 45. Envelope of a set (balanced) : I, p. 7. Envelope of a set (balanced convex) : II, p. 10. Envelope of a set (balanced convex closed) : II, p. 13 and p. 62. Envelope of a set (closed convex) : II, p. 13. Envelope of a set (convex) : II, p. 9. Envelope of a set (symmetric convex) : II, p. 10. Envelope of a set (symmetric convex closed) : II, p. 13. \mathscr{S}-equihypocontinuous, \mathfrak{T}-equihypocontinuous, (\mathscr{S}, \mathfrak{T})-equihypocontinuous set : III, p. 47, exerc. 7.

Exhaustion of a Hausdorff locally convex space, exhaustible space : III, p. 49, exerc. 1. Extremal generator of a convex cone : II, p. 57. Extremal point of a convex set : II, p. 54.

Facet : II, p. 87, exerc. 3. Facet (dual) : II, p. 87, exerc. 6. Family (orthonormal) : V, p. 21. Family (topologically independent) : I, p. 11. Filtered set of semi-norms : II, p. 3. Final locally convex topology : II, p. 27. Final subspace : V, p. 41. Fock spaces : V, p. 32 and p. 34. Form (bilinear) putting two spaces in duality : II, p. 40. Form (complex linear, real linear) : II, p. 61. Form (hermitian) : V, p. 1. Form (positive hermitian) : V, p. 2. Form (positive linear) : II, p. 13. Form (separating hermitian) associated with a hermitian form : V, p. 2. Fréchet space : II, p. 24 and p. 63. Function (concave, convex, strictly concave, strictly convex) : II, p. 16-17. Function (positive definite) : V, p. 8. Function (positively homogeneous), sub-linear function : II, p. 19-20. Fundamental system of semi-norms : II, p. 3.

Gauge of a convex set : II, p. 20. Gaussian space : V, p. 32. Generated (bornology) by a family of sets : III, p. 1. Generator (extremal) of a convex cone : II, p. 57. Gevrey’s spaces : III, p. 10. Gram’s determinant : V, p. 71, exerc. 7. Grothendieck’s theorem : III, p. 20. Group on which a mean can be defined : IV, p. 72, exerc. 4.

Haar’s theorem : II, p. 83, exerc. 8. Hadamard’s inequalities : V, p. 37. Hahn-Banach theorem : II, p. 22, p. 36 and p. 63. Half-spaces (closed, open) determined by a closed hyperplane : II, p. 15. Hardy space : V, p. 7. Hausdorff completion of a topological vector space : I, p. 6. Hausdorff vector space associated with a topological vector space : I, p. 4. Helly’s theorem : II, p. 68, exerc. 21. Hermitian endomorphism : V, p. 44. Hilbert-Schmidt mapping : V, p. 52. Hilbert space, hilbertian space : V, p. 6. Hyperplane (support) of a set : II, p. 37. \mathscr{S}-hypocontinuous, \mathscr{T}-hypocontinuous, (\mathscr{S}, \mathscr{T})-hypocontinuous bilinear mapping : III, p. 30.

Index of a Fredholm operator : IV, p. 66, exerc. 21. Induced prehilbertian structure on a vector subspace : V, p. 6. Inductive limit of locally convex spaces or topologies : II, p. 29. Inequalities (Hadamard’s) : V, p. 37. Inequality (Bessel’s) : V, p. 21. Inequality (Cauchy-Schwarz) : V, p. 3. Infra-barrelled space : III, p. 44, exerc. 7. Initial subspace : V, p. 41. Initial topology : IV, p. 4. Internal point of a convex set : II, p. 26.

James-Klee theorem : IV, p. 57, exerc. 25. R. C. James’ space : IV, p. 71, exerc. 18.

Krein-Milman theorem : II, p. 55. Krein’s theorem : IV, p. 37.

Legendre polynomial : V, p. 24. Locally convex complex space : II, p. 62. Locally convex real space : II, p. 23. Locally convex topology : II, p. 23 and p. 62.

Mackey theorem : IV, p. 2. Mackey topology : IV, p. 2. Mapping (Hilbert-Schmidt) : V, p. 52. Markoff-Kakutani theorem : IV, p. 39. Matrix (Hilbert’s) : V, p. 75, exerc. 3. Matrix with respect to orthonormal bases : V, p. 22. Mean : IV, p. 40. Metrisable topological vector space : I, p. 16. Minimal type (space of) : II, p. 85, exerc. 13. Montel space : IV, p. 18.

Normal endomorphism : V, p. 43.

Open half-spaces defined by a closed hyperplane : II, p. 15. Operator (Fredholm) : IV, p. 66, exerc. 21. Operator (unitary) : V, p. 41. Ordered vector space : II, p. 12. Orthogonal of a subspace for spaces in duality : II, p. 44. Orthogonal projector, orthoprojector : V, p. 13. Orthogonal sets in a prehilbertian space : V, p. 13. Orthogonal sets in spaces in duality : II, p. 41. Orthogonal to a subset in a prehilbertian space : V, p. 13. Orthogoanl vectors for a duality : II, p. 41. Orthogonal vectors in a prehilbertian space : V, p. 5. Orthonormal basis : V, p. 22. Orthonormal set, family : V, p. 21. Orthonormalisation : V, p. 24. Orthoprojector (initial, final) : V, p. 41.

Parabolic convex set : II, p. 67, exerc. 17. Parseval’s relation : V, p. 22. Partially isometric mapping : V, p. 42. Pointed cone : II, p. 10. Points (exposed, of strict convexity) : II, p. 88, exerc. 6. Points on the same side, strictly on the same side, of a hyperplane : II, p. 9. Polar decomposition of a Hilbert-Schmidt endomorphism : V, p. 79, exerc. 14. Polar of a set : II, p. 44 and 64. Polarization formulas : V, p. 2. Polyhedron : II, p. 90, exerc. 24. Positive endomorphism : V, p. 45. Positive hermitian form : V, p. 2. Positively homogeneous function : II, p. 19. Predual : IV, p. 56, exerc. 23. Prehilbertian semi-norm : V, p. 4. Prehilbertian space : V, p. 4. Preordered vector space : II, p. 12. Principle of condensation of singularities : III, p. 42, exerc. 10.

Product bornology : III, p. 2. Projection on a convex set : V, p. 11. Projection (orthogonal) : V, p. 13. Proper pointed convex cone : II, p. 11. Pythagoras’ theorem : V, p. 12.

Quasi-complete space : III, p. 8.

Real linear form, real linear variety : II, p. 61. Real locally convex space : II, p. 23. Reflexive space : IV, p. 16. Relatively bounded space.: III, p. 43, exerc. 6. Representation (unitary) : IV, p. 44. Ryll-Nardzewski theorem : IV, p. 43.

Scalar product : V, p. 5. Scalar square : V, p. 5. Segment (closed, open, open at x and closed at y) : II, p. 7. Semi-automorphism of prehilbertian spaces : V, p. 6. Semi-barrelled space : IV, p. 21. Semi-complete space : III, p. 7. Semi-norm : II, p. 1. Semi-normed space : II, p. 2. Semi-norm (prehilbertian) : V, p. 4. Semi-reflexive space : IV, p. 15. Separated completion of a prehilbertian space : V, p. 8. Separated completion of a topological vector space : I, p. 6. Separated (sets) by a closed hyperplane : II, p. 37. Separately continuous bilinear mapping : III, p. 28. Separately equicontinuous : III, p. 47. Separating duality : II, p. 41. Side of a hyperplane (points on the same, strictly on the same) : II, p. 8. Simplex : II, p. 71, exerc. 41. Šmulian theorem : IV, p. 36. Sobolev space : V, p. 6. Sole of a cone : II, p. 60. Space (DF) : IV, p. 57, exerc. 2. Space (topological vector) : I, p. 1. Space (weak) : II, p. 42. Square (scalar) : V, p. 5. Starshaped set : II, p. 65, exerc. 1. Strict inductive limit of a sequence of locally convex spaces, topologies : II, p. 33. Strictly concave, strictly convex function : II, p. 16-17. Strictly separated (sets) by a closed hyperplane : II, p. 37. Strong dual : III, p. 14. Strongly bounded subset of E′ : III, p. 14. Sub-linear function : II, p. 20. Subspace (final, initial) of a continuous linear mapping : V, p. 41. Subspace (prehilbertian) : V, p. 6. Sum (external hilbertian) of Hilbert spaces : V, p. 18. Sum (hilbertian) of vector subspaces : V, p. 18. Sum (topological direct) of locally convex spaces or topologies : II, p. 30. Support variety : II, p. 87, exerc. 3. Symmetric convex closed envelope of a set : II, p. 13.

Tchebycheffs theorem : II, p. 84, exerc. 8. Tensor product (hilbertian) : V, p. 28. Tensor product of prehilbertian spaces : V, p. 26-27.

Topologically independent set, family : I, p. 12 and p. 11. $\mathscr{S}$-topology : III, p. 13 and IV, p. 2. Topology compatible with an ordered vector space structure : II, p. 15. Topology compatible with a vector space structure : I, p. 1. Topology defined by a semi-norm, by a set of semi-norms : II, p. 2-3. Topology (initial) : IV, p. 4. Topology (locally convex) : II, p. 23 and p. 62. Topology (Mackey) : IV, p. 2. Topology of simple, compact, precompact, compact convex, bounded convergence : III, p. 14. Topology (weak) : II, p. 42. Topology (weakened) : IV, p. 4. Total set : I, p. 11. Trace of a continuous positive endomorphism : V, p. 49. Trace of a positive quadratic form with respect to another : V, p. 57. Transformation (affine) : IV, p. 39. Transpose of a continuous linear mapping : II, p. 46 and IV, p. 6.

Ultrabornological space : III, p. 45, exerc. 19. Ultranorm : I, p. 26, exerc. 12. Ultra-semi-norm : II, p. 2. Unitary operator : V, p. 41. Unitary representation : IV, p. 44. Unpointed cone : II, p. 10.

Variety (complex linear, real linear) : II, p. 61. Variety (support) : II, p. 87, exerc. 3.

Weak dual : III, p. 14. Weak topology, weak space : II, p. 42. Weakened topology : IV, p. 4. Weakly bounded : III, p. 14.