Linear Operators: General theory |
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Page 561
... matrices A in Em . Since e - tAtAI , the columns of the matrix et form a set of m linearly independent solutions of the equation dy / dt = Ay . ( d ) If T is such that ( 2 ) = 1 for λ e σ ( T ) , Theorem 8 yields k ƒ ( T ) — 21 ( 2 ...
... matrices A in Em . Since e - tAtAI , the columns of the matrix et form a set of m linearly independent solutions of the equation dy / dt = Ay . ( d ) If T is such that ( 2 ) = 1 for λ e σ ( T ) , Theorem 8 yields k ƒ ( T ) — 21 ( 2 ...
Page 565
... matrix of dY / dt = A ( t ) Y which is non - singular . Show that the set of all non - singular matrix solutions are precisely the matrices Y ( t ) C where C is any n × n constant , non- singular matrix . - 26 Let A ( t ) have period p ...
... matrix of dY / dt = A ( t ) Y which is non - singular . Show that the set of all non - singular matrix solutions are precisely the matrices Y ( t ) C where C is any n × n constant , non- singular matrix . - 26 Let A ( t ) have period p ...
Page 607
... matrix were used almost from the beginning of the theory , and by 1867 Laguerre [ 1 ] had considered infinite power series in a matrix in constructing the exponential function of a matrix . Sylvester [ 1 , 2 ] constructed arbitrary ...
... matrix were used almost from the beginning of the theory , and by 1867 Laguerre [ 1 ] had considered infinite power series in a matrix in constructing the exponential function of a matrix . Sylvester [ 1 , 2 ] constructed arbitrary ...
Contents
A Settheoretic Preliminaries | 1 |
10 | 30 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra analytic arbitrary B-space B₁ ba(S Banach Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense E₁ element equation equivalent exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue measure Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space neighborhood non-negative non-zero normed linear space o-field o-finite open set operator topology positive measure space properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact X₁ zero ΕΕΣ