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 Elements of Set Theory 183 1 Basic operations on sets 6 Relations 12 Functions 20 Order 29 Some Properties of real numbers 34 The axiom of choice and its equivalents 35 Cardinality and denumerability 43
 Linear transformations or Homomorphism of linear spaces 182 Properties of linear transformations 183 IS Some particular linear transformations 184 Rank and nullity of a linear transformation 186 Linear transformations as vectors 187 Product of linear transformations 189 Algebra or linear algebra 192 Polynomials 193

 Decimal Ternary and binary representations 53 CardinalArithmetic 54 Cantors ternary set 70 Order types and ordinal numbers 73 Metric Spaces 84138 84 Euclidean spaces 89 Some important inequalities 92 Solved Examples 93 Bounded and unbounded Metric spaces 101 Spheres or balls 102 Open Sets 105 Closed Sets 108 Neighbourhoods 111 Adherent points 112 Closure interior exterior and boundary of a set 114 Subspaces 117 Dense and nondense sets separable spaces 119 Sequences and subsequences in a metric space 121 Cauchy sequences 123 Complete metric spaces 124 Completeness and contracting mappings 128 Some complete metric spaces 129 Completion of a metric space 136 Linear Spaces 139209 139 General properties of linear spaces 146 Linear subspaces 147 Algebra of subspaces 149 Linear combination of vectors Linear span of a set 150 Linear sum of two subspaces 151 Direct sum of spaces 152 Quotient space 155 Linear dependence and linea independence of vectors 163 Hamel Basis of linear space 166 Dimension of a linear space 170 Isomorphism of linear spaces 172
 Invertible or nonsingular linear transformations 194 Linear functionals 197 Linear functionals in finitedimensional spaces 198 Extension theorems for linear transformations 200 Reflexivity 203 Projections 206 CHAPTER 4 210 The classical Banach space Lp 235 Subspaces and Quotient spaces of Banach spaces 241 Continuous linear transformations 249 Equivalent norms 263 Ricszlemma 269 Convexity 270 Linear functionals and the Hahn Banach theorem 273 The natural imbedding of N into A Reflexivity 287 The open mapping theorem 295 Projections on Banach spaces 300 The Closed graph theorem 301 Uniform bounded principle 305 The conjugate of an operator 307 Hilbert Spaces 316398 316 Hilbert spaces 317 Some properties of Hilbert 318 Orthogonal complements 330 Orthonormal sets 336 The Conjugate space H 353 The Adjoint of an operator 361 Self adjoint operators 367 Normal and Unitary operators 373 Perpendicular Projections 384 CHAPTER 6 399 Matrix of a Linear transformation 401 The spectral thumcm 415 Index ivi 431 ÕŞÊŞ «·‰‘—