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Page 152
... ɛ K ) , h ( t ) < f ( t ) + ε ( t & K ) . whereas ( 70 ) and ( 71 ) imply ( 73 ) Finally , ( 69 ) follows from ( 72 ) ... ƒ must also belong to a ; ƒ is defined by ƒ ( x ) = f ( x ) . 7.31 . Theorem . Suppose a is a self - adjoint algebra ...
... ɛ K ) , h ( t ) < f ( t ) + ε ( t & K ) . whereas ( 70 ) and ( 71 ) imply ( 73 ) Finally , ( 69 ) follows from ( 72 ) ... ƒ must also belong to a ; ƒ is defined by ƒ ( x ) = f ( x ) . 7.31 . Theorem . Suppose a is a self - adjoint algebra ...
Page 241
Walter Rudin. the integral of ƒ over E is defined , although ƒ is not integrable in the above sense of the word ; ƒ ... ɛ E , and μ ( E ) < + ∞ , then aμ ( E ) ≤ [ gƒ du ≤ bμ ( E ) . E ( c ) If ƒ and g ɛ £ ( μ ) on E , and if f ( x ) ...
Walter Rudin. the integral of ƒ over E is defined , although ƒ is not integrable in the above sense of the word ; ƒ ... ɛ E , and μ ( E ) < + ∞ , then aμ ( E ) ≤ [ gƒ du ≤ bμ ( E ) . E ( c ) If ƒ and g ɛ £ ( μ ) on E , and if f ( x ) ...
Page 251
... ƒ . 10.35 . Theorem . Suppose fε L2 ( μ ) and g ɛ L2 ( μ ) . Then fg & L ( μ ) , and ( 97 ) √x | fg | du ≤ || f ... ƒ || 2 + 2x √x | fg | dμ + X2 || g || 2 , which holds for every real X. 10.36 . Theorem . Iƒ ƒ ɛ L2 ( μ ) and g ɛ L2 ...
... ƒ . 10.35 . Theorem . Suppose fε L2 ( μ ) and g ɛ L2 ( μ ) . Then fg & L ( μ ) , and ( 97 ) √x | fg | du ≤ || f ... ƒ || 2 + 2x √x | fg | dμ + X2 || g || 2 , which holds for every real X. 10.36 . Theorem . Iƒ ƒ ɛ L2 ( μ ) and g ɛ L2 ...
Contents
Preface | 1 |
ELEMENTS OF SET THEORY | 21 |
NUMERICAL SEQUENCES AND SERIES | 41 |
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a₁ B₂ bounded variation C'-mapping called Cauchy sequence choose closed complex numbers continuous function continuous on a,b converges uniformly Corollary countable set Definition denote diverges Example Exercise exists f is differentiable finite fn(x follows Fourier series function defined function f ƒ and g ƒ ɛ ƒ is continuous given Hence Hint holds implies inequality infinite integer integral interval k-form Lebesgue Let f lim inf lim sup limit point linear mean value theorem measurable functions metric space monotonic functions neighborhood nonnegative notation number system obtain open set partial sums partition polynomials positive integer power series properties rational numbers real function real numbers Riemann integral say that ƒ shows Suppose f Theorem uniform convergence uniformly continuous upper number variation on a,b vector space y₁ Σ Σ