Linear Operators: General theory |
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Page 240
... scalar function f on S is E - measurable if f - 1 ( A ) for every Borel set A in the range of f . It is clear that ... scalar functions on S. The norm is given by = sup f ( s ) . 8ES 14. The space C ( S ) is defined for a topological ...
... scalar function f on S is E - measurable if f - 1 ( A ) for every Borel set A in the range of f . It is clear that ... scalar functions on S. The norm is given by = sup f ( s ) . 8ES 14. The space C ( S ) is defined for a topological ...
Page 256
... scalar product ( iv ) ( [ x1 , ... , xn ] , [ Y1 , ... , Yn ] ) = [ Y1 , · · n Σ ( x , y ) i i = 1 where ( , ) , is the scalar product in X. Thus the norm in a direct sum of Hilbert spaces is always given by ( iii ) . To summarize , we ...
... scalar product ( iv ) ( [ x1 , ... , xn ] , [ Y1 , ... , Yn ] ) = [ Y1 , · · n Σ ( x , y ) i i = 1 where ( , ) , is the scalar product in X. Thus the norm in a direct sum of Hilbert spaces is always given by ( iii ) . To summarize , we ...
Page 323
... scalar valued and u - integrable , the integral of f with respect to u over E is an unambiguously defined element of X ; ( b ) if ƒ and g are scalar valued and u - integrable , if a and ẞ are sca- lars , and if Ee Z , then √g { aƒ ( s ) ...
... scalar valued and u - integrable , the integral of f with respect to u over E is an unambiguously defined element of X ; ( b ) if ƒ and g are scalar valued and u - integrable , if a and ẞ are sca- lars , and if Ee Z , then √g { aƒ ( s ) ...
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