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CHAPTER Ⅰ．LINEAR MANIFOLDS AND LINEAR TRANSFORMATIONS IN Bn 1

1．Vectors and operations on vectors 1

2．Linear manifolds 8

3．Projections;orthonormal bases of a L．M．;complete orthonormal systems 12

4．Products and sums of L．M．＇s 17

5．Orthogonal complements;systems of homogeneous linear equations 21

6．Linear transformations;systems of non-homogeneous linear equations 24

7．Sums and products of L．T．＇s 34

CHAPTER Ⅱ．SPECIAL LINEAR TRANSFORMATIONS IN Bn 40

8．Hermitian transformations;eigen-values，eigen-manifolds and reduction 40

9．Normal transformations and unitary transformations 45

10．Projectors 49

CHAPTER Ⅲ．SPECTRAL REPRESENTATION OF HERMITIAN TRANSFORMATIONS IN Bn 58

11．Continuous functions in Bn 58

12．Orthonormal systems of eigen-solutions of a H．T．The spectral representations 60

13．Unitary transformations of a Hermitian form to principal axes 67

14．Further maximal properties of the eigen-values．inequalities 73

15．Functional calculus of H．T．＇s．The resolvent and its properties 78

16．Transformations commutative with H．T．＇s．Normal transformations 86

CHAPTER Ⅳ．SPECTRAL PROPERTIES OF LINEAR TRANSFORMATIONS IN Bn 97

17．Eigen-values and principal manifolds 97

18．The minimal polynomial 103

19．Functional calculus for L．T．＇s．The resolvent 108

20．Canonical bases of a principal manifold 117

21．Jordan＇s canonical form for a nilpotent L．T．Commutativity 122

22．Jordan＇s canonical form for any L．T．;elementary divisors and Segre characteristic;similar transformations 130

23．Commutativity of L．T．＇s 138

CHAPTER Ⅴ．VECTOR SPACES WITH POSITIVE HERMITIAN METRIC FORMS 151

24．The spaces B＇n and Bn0 151

25．Hermitian forms in B＇n and Bn0 and the pencil H－λG in Bn 159

26．L．T．＇s that are Hermitian or normal in some B＇n 164

27．Application to the dynamical theory of small oscillations 166

Notes 170

References 184

Index of Authors 190

General Index 191