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Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter3: Functions And Graphs

Section3.2: Graphs Of Equations

Problem 76E

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Question

For X and Y normed linear spaces, let {T} be a sequence in L(X, Y) such that T →T in L(X, Y)
and let {n} be a sequence in X such that un → u in X.
Vn EN.
Let = 1 in the definition of convergence of {T} to T in L(X, Y). Show that ||Tn|| ≤ M,
where M = sup{||T₁||, ||T₂||, ..., ||TN-1||, 1 + ||T||}, for some N € N.

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Prove the conjecture made in the previous exercise.
If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local extremum offon (a,c) ?
a) Let (h,), (tn) be sequences of bounded functions on A that converges uniformly on A to h, t respectively. Show that (h, tn) converges uniformly on A to ht.
(a) If (xn) is a Cauchy sequence in the discrete metric space (M, d), then prove that for some NEZ*, we get: ..... (b) Prove that the sequence (n) is Cauchy sequence in (R,d), where d(x,y) = 2-; vx,y eR*. On the other hand, show that (n) is not convergent in (R*,d). (c) If (xn) is a Cauchy sequence in (R,I), then prove that ((xn , sin( xn ))) is Cauchy sequence in (R2,).
Lemma 2.42. If (x,) converges weakly to x in a normed linear space X, then every subsequence of (x„) converges weakly to x.
For X and Y normed linear spaces, let {T} be a sequence in L(X, Y) such that Tn → T in L(X,Y) and let {n} be a sequence in X such that un → u in X. Let ε = 1 in the definition of convergence of {T} to T in L(X, Y). Show that for all n ≤ N, ||Tn(Un) – T(u)||y ≤ ||Tn|| · ||Un − U||x + ||Tn − T|| · ||u||x.
(a) If (xn) is a Cauchy sequence in the discrete metric space (M,d), then prove that for some NEZ*, we get: (xn) = (x1,X2,...,XN-1, XN,XN ,XN.). (b) Prove that the sequence (n) is Cauchy scquence in (R*,d), where d(x,y) = 2-: vx,y ER*. On the other hand, show that (n) is not convergent in (R,d). (c) If (xn) is a Cauchy sequence in (R,I), then prove that ((xn,sin( x, ))) is Cauchy sequence in (R2, l).

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