The following conditions are necessary and sufficient

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The following conditions are necessary and sufficient for the a ij to con- stitute a summation method:- (i) There exists a K such that X j =1 | a ij | ≤ K for all i. (ii) X j =1 a ij 1 as i → ∞ . (iii) a ij 0 as i → ∞ for each j . 9
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Exercise 26. Ces` aro’s summation method takes a sequence c 0 , c 1 , c 2 , . . . and replaces it with a new sequence whose n th term b n = c 1 + c 2 + · · · + c n n is the average of the first n terms of the old sequence. (i) By rewriting the statement above along the lines of Lemma 25 show that if the old sequence converges to c so does the new one. (ii) Examine what happens when c j = ( - 1) j . Examine what happens if c j = ( - 1) k when 2 k j < 2 k +1 . (iii) Show that, in the notation of Lemma 25, taking a n, 2 n = 1 , a n,m = 0 , otherwise, gives a summation method. Show that taking a n, 2 n +1 = 1 , a n,m = 0 , otherwise, also gives a summation method. Show that the two methods disagree when presented with the sequence 1 , - 1 , 1 , - 1 , . . . . Another important consequence of the Baire category theorem is the open mapping theorem. (Recall that a complete normed space is called a Banach space.) Theorem 27 (Open mapping theorem). Let E and F be Banach spaces and T : E F be a continuous linear surjection. Then T is an open map (that is to say, if U is open in E we have TU open in F .) This has an immediate corollary. Theorem 28 (Inverse mapping theorem). Let E and F be Banach spaces and let T : E F be a continuous linear bijection. Then T - 1 is continuous. The next exercise is simple, and if you can not do it this reveals a gap in your knowledge (which can be remedied by asking the lecturer) rather that in intelligence. Exercise 29. Let ( X, d ) and ( Y, ρ ) be metric spaces with associated topologies τ and σ . Then the product topology induced on X × Y by τ and σ is the same as the topology given by the metric 4 (( x 1 , y 1 ) , ( x 2 , y 2 )) = d ( x 1 , x 2 ) + ρ ( y 1 , y 2 ) . The inverse mapping theorem has the following useful consequence. Theorem 30 (Closed graph theorem). Let E and F be Banach spaces and let T : E F be linear. Then T is continuous if and only the graph { ( x, Tx ) : x E } is closed in E × F with the product topology. 10
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5 Zorn’s lemma and Tychonov’s theorem Let A be a non empty set and, for each α A , let X α be a non-empty set. Is Q α A X α non-empty (or, equivalently, does there exist a function f : A S α A X α with f ( α ) X α )? It is known that the standard axioms of set theory do not suffice to answer this question in general. (In particular cases they do suffice. If X α = A for all α A then f ( α ) = α will do.) Specifically, if there exists any model for standard set theory, then there exist models for set theory obeying the standard axioms in which the answer to our question is always yes (such systems are said to obey the axiom of choice) and there exist models in which the answer is sometimes no.
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  • Fall '08
  • Groah
  • Math, Compact space, Banach space, Banach, Banach algebra, commutative Banach

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