Linear Operators: General theory |
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Page 45
... matrix ( a ;; ) . If T is a linear operator , then it can be shown that the determinants of the matrices of T in terms of any two bases are equal , and so we may and shall call their common value the determinant of T , denoted by det ...
... matrix ( a ;; ) . If T is a linear operator , then it can be shown that the determinants of the matrices of T in terms of any two bases are equal , and so we may and shall call their common value the determinant of T , denoted by det ...
Page 564
... matrix differential equation dY dt = A ( t ) Y where a solution is a nxn complex valued matrix Y ( t ) , differentiable and satisfying the differential equation for each te I. Show that for each to € I and each complex ( constant ) matrix ...
... matrix differential equation dY dt = A ( t ) Y where a solution is a nxn complex valued matrix Y ( t ) , differentiable and satisfying the differential equation for each te I. Show that for each to € I and each complex ( constant ) matrix ...
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 nxn constant , non- singular matrix . = 26 Let A ( t ) have period p > 0 ...
... 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 nxn constant , non- singular matrix . = 26 Let A ( t ) have period p > 0 ...
Contents
8 | 28 |
Algebraic Preliminaries | 34 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra Amer analytic arbitrary B-space ba(S Banach Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense differential disjoint Doklady Akad E₁ element equation exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism L₁ L₁(S Lebesgue Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ