Topological Vector Spaces II: (237 Grundlehren der Mathematischen Wissenschaften)

In the preface to Volume One I promised a second volume which would contain the theory of linear mappings and special classes of spaces im- portant in analysis. It took me nearly twenty years to fulfill this promise, at least to some extent. To the six chapters of Volume One I added two new chapters, one on linear mappings and duality (Chapter Seven), the second on spaces of linear mappings (Chapter Eight). A glance at the Contents and the short introductions to the two new chapters will give a fair impression of the material included in this volume. I regret that I had to give up my intention to write a third chapter on nuclear spaces. It seemed impossible to include the recent deep results in this field without creating a great further delay. A substantial part of this book grew out of lectures I held at the Mathematics Department of the University of Maryland* during the academic years 1963-1964, 1967-1968, and 1971-1972. I would like to express my gratitude to my colleagues J. BRACE, S. GOLDBERG, J. HORVATH, and G. MALTESE for many stimulating and helpful discussions during these years. I am particularly indebted to H. JARCHOW (Ziirich) and D. KEIM (Frankfurt) for many suggestions and corrections. Both have read the whole manuscript. N. ADASCH (Frankfurt), V. EBERHARDT (Miinchen), H. MEISE (Diisseldorf), and R. HOLLSTEIN (Paderborn) helped with important observations.

Table of Contents:
of Vol. II.- Seven Linear Mappings and Duality.- 32. Homomorphisms of locally convex spaces.- 1. Weak continuity.- 2. Continuity.- 3. Weak homomorphisms.- 4. The homomorphism theorem.- 5. Further results on homomorphisms.- 33. Linear continuous mappings of (B)-and (F)-spaces.- 1. First results in normed spaces.- 2. Metrizable locally convex spaces.- 3. Applications of the Banach-Dieudonne theorem.- 4. Homomorphisms in (B)- and (F)-spaces.- 5. Separability. A theorem of Sobczyk.- 6. (FM)-spaces.- 34. The theory of Ptak.- 1. Nearly open mappings.- 2. Ptak spaces and the Banach-Schauder theorem.- 3. Some results on Ptak spaces.- 4. A theorem of Kelley.- 5. Closed linear mappings.- 6. Nearly continuous mappings and the closed-graph theorem.- 7. Some consequences, the Hellinger-Toeplitz theorem.- 8. The theorems of A. and W. Robertson.- 9. The closed-graph theorem of Komura.- 10. The open mapping theorem of Adasch.- 11. Kalton's closed-graph theorems.- 35. De Wilde's theory.- 1. Webs in locally convex spaces.- 2. The closed-graph theorems of De Wilde.- 3. The corresponding open-mapping theorems.- 4. Hereditary properties of webbed and strictly webbed spaces.- 5. A generalization of the open-mapping theorem.- 6. The localization theorem for strictly webbed spaces.- 7. Ultrabornological spaces and fast convergence.- 8. The associated ultrabornological space.- 9. Infra-(u)-spaces.- 10. Further results.- 36. Arbitrary linear mappings.- 1. The singularity of a linear mapping.- 2. Some examples.- 3. The adjoint mapping.- 4. The contraction of A.- 5. The adjoint of the contraction.- 6. The second adjoint.- 7. Maximal mappings.- 8. Dense maximal mappings.- 37. The graph topology. Open mappings.- 1. The graph topology.- 2. The adjoint of AIA.- 3. Nearly open mappings.- 4. Open mappings.- 5. Ptak spaces. Open mapping theorems.- 6. Linear mappings in metrizable spaces.- 7. Open mappings in (B)- and (F)-spaces.- 8. Domains and ranges of closed mappings of (F)-spaces.- 38. Linear equations and inverse mappings.- 1. Solvability conditions.- 2. Continuous left and right inverses.- 3. Extension and lifting properties.- 4. Inverse mappings.- 5. Solvable pairs of mappings.- 6. Infinite systems of linear equations.- Eight Spaces of Linear and Bilinear Mappings.- 39. Spaces of linear mappings.- 1. Topologies on L (E, F).- 2. The Banach-Mackey theorem.- 3. Equicontinuous sets.- 4. Weak compactness. Metrizability.- 5. The Banach-Steinhaus theorem.- 6. Completeness.- 7. The dual of Ls (E, F).- 8. Some structure theorems.- 40. Bilinear mappings.- 1. Fundamental notions.- 2. Continuity theorems for bilinear maps.- 3. Extensions of bilinear mappings.- 4. Locally convex spaces of bilinear mappings.- 5. Applications. Locally convex algebras.- 41. Projective tensor products of locally convex spaces.- 1. Some complements on tensor products.- 2. The projective tensor product.- 3. The dual space. Representations of E ???F.- 4. The projective tensor product of metrizable and of (DF)-spaces.- 5. Tensor products of linear maps.- 6. Further hereditary properties.- 7. Some special cases.- 42. Compact and nuclear mappings.- 1. Compact linear mappings.- 2. Weakly compact linear mappings.- 3. Completely continuous mappings. Examples.- 4. Compact mappings in Hilbert space.- 5. Nuclear mappings.- 6. Examples of nuclear mappings.- 7. The trace.- 8. Factorization of compact mappings.- 9. Fixed points and invariant subspaces.- 43. The approximation property.- 1. Some basic results.- 2. The canonical map of E ???F in B (E?s x F?s).- 3. Another interpretation of the approximation property.- 4. Hereditary properties.- 5. Bases, Schauder bases, weak bases.- 6. The basis problem.- 7. Some function spaces with the approximation property.- 8. The bounded approximation property.- 9. Johnson's universal space.- 44. The injective tensor product and the ?-product.- 1. Compatible topologies on E ? F.- 2. The injective tensor product.- 3. Relatively compact subsets of E?F and E ???F.- 4. Tensor products of mappings.- 5. Hereditary properties.- 6. Further results on tensor product mappings.- 7. Vector valued continuous functions.- 8. ?-tensor product with a sequence space.- 45. Duality of tensor products.- 1. First results.- 2. A theorem of Schatten.- 3. Buchwalter's results on duality.- 4. Canonical representations of integral bilinear forms.- 5. Integral mappings.- 6. Nuclear and integral norms.- 7. When is every integral mapping nuclear?.- Author and Subject Index.