| Linear Topological Spaces | p. 1 |
| Topological spaces and metric spaces | p. 1 |
| Linear topological spaces | p. 3 |
| Countably normed spaces | p. 6 |
| Continuous linear functionals | p. 9 |
| Weak and strong topologies | p. 11 |
| Perfect spaces | p. 15 |
| Linear operators | p. 18 |
| Inductive limits and unions of topological spaces | p. 20 |
| Problems | p. 24 |
| Spaces of Generalized Functions | p. 25 |
| Fundamental spaces and generalized functions | p. 25 |
| The spaces K {M[subscript b]} | p. 29 |
| The spaces Z{M[subscript p]} | p. 33 |
| Multiplication and the derivatives of generalized functions | p. 34 |
| Structure of generalized functions on K{M[subscript p]} | p. 37 |
| Problems | p. 41 |
| Theory of Distributions | p. 43 |
| Spaces of functions | p. 43 |
| Partition of unity | p. 45 |
| Definition and some properties of distributions | p. 47 |
| Derivatives of distributions | p. 52 |
| Structure of distributions | p. 59 |
| Structure of distributions (continued) | p. 63 |
| Distributions having support on compact sets or on subspaces | p. 68 |
| Tensor product of distributions | p. 70 |
| Product of distributions by functions. Applications to differential equations | p. 73 |
| Convolutions of distributions | p. 76 |
| Convolutions of distributions with smooth functions | p. 79 |
| The spaces K[subscript r]{M[subscript p]}, {D[subscript L superscript r]} and the structure of their generalized functions | p. 83 |
| Convolution equations | p. 88 |
| The spaces (S) and (S') | p. 90 |
| Fourier transforms of distributions | p. 93 |
| Problems | p. 98 |
| Convolutions and Fourier Transforms of Generalized Functions | p. 101 |
| Fourier transforms of fundamental functions | p. 101 |
| Fourier transforms of generalized functions | p. 103 |
| Convolutions of generalized functions | p. 104 |
| The convolution theorems | p. 110 |
| Problems | p. 112 |
| W Spaces | p. 113 |
| Theorems on complex analytic functions | p. 113 |
| Definition of W spaces | p. 125 |
| Operators in W spaces | p. 128 |
| Fourier transforms of W spaces | p. 131 |
| Nontriviality and richness of W spaces | p. 136 |
| Problems | p. 139 |
| Fourier Transforms of Entire Functions | p. 140 |
| Entire functions of order [less than or equal] p and of fast decrease | p. 140 |
| Entire functions of order [less than or equal] 1 | p. 142 |
| Entire functions of order [less than or equal] 1 and of slow increase | p. 145 |
| Entire functions of order [less than or equal] p and of slow increase | p. 147 |
| Entire functions of order [less than or equal] p and of mildly fast increase | p. 150 |
| Entire functions of order [less than or equal] p and of fast increase | p. 158 |
| Proof of Lemma 2 | p. 159 |
| Problems | p. 163 |
| The Cauchy Problem for Systems of Partial Differential Equations | p. 164 |
| Systems of partial differential equations and the Cauchy problem | p. 164 |
| Auxiliary theorems on functions of matrices | p. 168 |
| Uniqueness of solutions of the Cauchy problem | p. 177 |
| Existence of generalized solutions | p. 184 |
| Lemmas on convolutions | p. 186 |
| Existence theorems for parabolic systems | p. 191 |
| An existence theorem for hyperbolic systems | p. 196 |
| Existence theorems for correctly posed systems | p. 198 |
| Existence theorems for mildly incorrectly posed systems | p. 200 |
| An existence theorem for incorrectly posed systems | p. 205 |
| Nonhomogeneous systems with time-dependent coefficients | p. 206 |
| Systems of convolution equations | p. 208 |
| Difference-differential equations | p. 210 |
| Inverse theorems | p. 218 |
| Proof of the Seidenberg-Tarski theorem | p. 225 |
| Problems | p. 235 |
| The Cauchy Problem in Several Time Variables | p. 237 |
| Uniqueness and existence of generalized solutions | p. 237 |
| Sobolev's lemma | p. 241 |
| Proof of Theorem 2 | p. 243 |
| Existence of classical solutions | p. 248 |
| The Goursat problem | p. 250 |
| Problems | p. 257 |
| S Spaces | p. 258 |
| Definition of S spaces | p. 258 |
| Operators in S spaces | p. 264 |
| Fourier transforms of S spaces | p. 266 |
| Nontriviality and richness of S spaces | p. 270 |
| Problems | p. 272 |
| Further Applications to Partial Differential Equations | p. 273 |
| A Phragmen-Lindelof type theorem | p. 273 |
| A Liouville type theorem | p. 279 |
| Fundamental solutions of equations with constant coefficients | p. 285 |
| Special distributions and Radon's problem | p. 289 |
| Fundamental solutions for hyperbolic equations | p. 295 |
| Problems | p. 299 |
| Differentiability of Solutions of Partial Differential Equations | p. 301 |
| Hypoelliptic equations and their fundamental solutions | p. 301 |
| Conditions for hypoellipticity | p. 304 |
| Conditions for hypoellipticity (continued) | p. 313 |
| Examples of hypoelliptic equations | p. 316 |
| Nonhomogeneous equations | p. 320 |
| Problems | p. 325 |
| Bibliographical Remarks | p. 326 |
| Bibliography | p. 329 |
| Index for Spaces | p. 333 |
| Index | p. 335 |
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