Principles of Mathematical Analysis |
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Page 30
... open if and only if its complement is closed . Proof : First , suppose E is closed . Choose x ε E. Then x E , and x ... sets , U Ga is open . ( b ) For any collection { Fa } of closed sets , Fa is closed . ( c ) For any finite collection ...
... open if and only if its complement is closed . Proof : First , suppose E is closed . Choose x ε E. Then x E , and x ... sets , U Ga is open . ( b ) For any collection { Fa } of closed sets , Fa is closed . ( c ) For any finite collection ...
Page 87
... sets in a metric space X , K is compact , F is closed . Prove that there exists > 0 such that d ( p , q ) > d if pɛ ... open and disjoint , and that A C V , B C W. ( Thus pairs of disjoint closed sets in a metric space can be covered by ...
... sets in a metric space X , K is compact , F is closed . Prove that there exists > 0 such that d ( p , q ) > d if pɛ ... open and disjoint , and that A C V , B C W. ( Thus pairs of disjoint closed sets in a metric space can be covered by ...
Page 235
... open intervals . To see this , it is sufficient to construct a countable base whose members are open intervals . By taking complements , it follows that every closed set is in M ( μ ) . ( b ) If A ε M ( u ) and e > 0 , there exist sets ...
... open intervals . To see this , it is sufficient to construct a countable base whose members are open intervals . By taking complements , it follows that every closed set is in M ( μ ) . ( b ) If A ε M ( u ) and e > 0 , there exist sets ...
Contents
Preface V | 1 |
ELEMENTS OF SET THEORY | 21 |
NUMERICAL SEQUENCES AND SERIES | 41 |
Copyright | |
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a₁ B₂ bounded variation C'-mapping called Cauchy sequence choose 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 integer integral interval k-form Lebesgue Lebesgue integral Let f lim inf lim sup limit point linear measurable functions metric space monotonic functions neighborhood nonnegative notation number system obtain open set P₁ partial sums partition polynomials positive integer properties Prove rational numbers real function real numbers Riemann integral say that f shows Suppose f Theorem uniform convergence uniformly continuous upper number variation on a,b vector space y₁