Introduction to the theory of partially ordered spaces

Title Page......Page 3
Copyright Information......Page 4
Contents......Page 5
Foreword......Page 11
1. General concept of partially ordered set......Page 17
2. Nets and directions......Page 19
3. Limit of a numerical direction......Page 22
4. Limit of a direction in a topological space......Page 24
5. Maximal and minimal elements. Zorn's lemma......Page 25
6. Suprema and infima......Page 27
7. Isomorphism of partially ordered sets......Page 29
1. General notions about lattices......Page 30
2. Sublattices......Page 32
3. Complete and Dedekind complete lattices......Page 33
4. Distributive lattices......Page 37
5. Boolean algebras......Page 40
6. Order convergence......Page 42
7. Order topology in lattices......Page 49
8. Some applications of lattices in topology......Page 51
9. Representation of a Boolean algebra in the form of a ring of open-closed sets......Page 53
1. Definition of a linear lattice......Page 60
2. Another approach to the definition of a linear lattice the concept of a Riesz space......Page 62
3. Examples of Riesz spaces......Page 66
4. Representation of the elements of a linear lattice in the form of the difference of positive elements. Absolute value of an element......Page 68
5. Distributivity of a linear lattice......Page 70
6. Disjoint elements and disjoint sets......Page 72
7. (o)-convergence in Riesz spaces......Page 76
9. Ideals......Page 79
10. The axiom of Archimedes and consequences of it......Page 82
11. Convergence with respect to a regulator......Page 83
12. Riesz spaces with unit......Page 85
13. Discrete elements......Page 89
14. Finite-dimensional Riesz spaces......Page 90
1. Definition of a Dedekind complete Riesz space and of a Dedekind o-complete Riesz space and their simplest properties......Page 93
2. Dedekind o-complete Riesz spaces with unit......Page 97
3. Projection on a band......Page 98
4. Properties of the projection operator......Page 103
5. Decomposition of a Dedekind o-complete Riesz space into bands......Page 104
6. Union of Dedekind complete Riesz spaces......Page 108
7. Decomposition of a Dedekind complete Riesz space into bands with unit and embedding of an arbitrary Dedekind complete Riesz space in a Dedekind complete Riesz space with unit......Page 111
8. Trace of an element in Dedekind o-complete Riesz spaces with unit......Page 112
9. Series in Dedekind o-complete Riesz spaces......Page 116
10. Integral representation of elements......Page 118
11. Completion of an Archimedean Riesz space to a Dedekind complete Riesz space......Page 124
12. Discrete Dedekind complete Riesz spaces......Page 129
1. Semi-continuous functions......Page 133
2. Dedekind complete Riesz space of continuous functions on a compactum......Page 138
3. Representation of the Dedekind complete Riesz space of bounded elements with the aid of continuous functions......Page 143
4. Embedding of an arbitrary Dedekind complete Riesz space in a Dedekind complete Riesz space of continuous functions......Page 149
5. Extended Dedekind a-complete Riesz spaces......Page 156
6. Maximal extension of a Dedekind complete Riesz space......Page 158
7. Representation of Archimedean Riesz spaces. Theorem on the preservation of relations......Page 161
8. Partially ordered rings......Page 162
1. Boolean algebras of countable type......Page 170
2. Dedekind complete Riesz spaces of countable type......Page 172
3. (o)-topology in Dedekind complete Riesz spaces of countable type......Page 174
4. Almost regular Dedekind complete Riesz spaces......Page 176
5. Regular Dedekind complete Riesz spaces......Page 177
6. Extended regular Dedekind complete Riesz spaces......Page 181
1. Normed lattices......Page 189
2. Banach lattices......Page 193
3. Dedekind complete normed lattices and Dedekind a-complete normed lattices......Page 194
4. Normability of a Riesz space of bounded elements......Page 195
5. Representation of a normed lattice of bounded elements with the aid of continuous functions......Page 198
6. KB-spaces......Page 204
7. KB-spaces with additive norm......Page 211
8. Countably-normed lattices......Page 213
1. Fundamental definitions......Page 220
2. Dedekind complete Riesz space of regular operators......Page 223
3. (o)-linear operators......Page 230
4. Completely linear operators......Page 235
5. Normal operators and a-normal operators......Page 237
6. Linear operators in normed lattices......Page 240
7. Linear operators in KB-spaces......Page 244
8. Linear operators in countably-normed lattices......Page 248
9. Additive functions on a Boolean algebra......Page 249
10. Integral representation of linear operators......Page 253
1. Regular functionals......Page 261
2. (o)-linear functionals and completely linear functionals......Page 263
3. (o)-linear functionals in the Dedekind complete Riesz space of bounded elements......Page 267
4. Linear functionals in normed lattices and in countably-normed lattices......Page 271
5. Embedding of a Riesz space in the second conjugate Dedekind complete Riesz space......Page 277
6. Reflexive Dedekind complete Riesz spaces......Page 283
7. Embedding of a normed lattice in the second (b)-conjugate space......Page 285
1. Extension of (bo)-linear operators......Page 288
2. Extension of (b)-linear operators......Page 291
3. Extension of positive operators......Page 293
4. Extension of regular operators from an Archimedean Riesz space to its Dedekind completion......Page 294
5. Extension of some (o)-linear functionals......Page 297
6. Applications to the theory of the integral......Page 307
1. Fundamental information about self-adjoint operators......Page 311
2. Dedekind complete Riesz spaces of bounded self-adjoint operators......Page 314
3. Existence of the principal unit in a strongly closed ring of bounded self-adjoint operators......Page 321
4. Spectral decomposition of self-adjoint operators......Page 324
5. Boolean measure on the real line......Page 328
6. Condition of countable additivity of a Boolean measure......Page 336
7. Measurable functions of elements of a Dedekind complete Riesz space......Page 339
8. Functions of a bounded self-adjoint operator......Page 346
1. Lattice-normed spaces......Page 350
2. General theorems on the method of successive approximations......Page 353
3. Application of the general theorems to linear equations......Page 357
4. Derivative of an operator......Page 365
5. Newton's method......Page 370
6. Monotonic processes of successive approximations......Page 373
1. Krein spaces......Page 378
2. Positive (b)-linear functionals in Krein spaces......Page 382
3. Krein lattices and their representation......Page 388
4. On fixed points of adjoint operators......Page 390
Literature......Page 395
Index......Page 401