For X and Y normed linear spaces, let {Tn} be a sequence in L(X,Y) such that Tn →T in L(X,Y)and let {un} be a sequence in X such that un → u in X.Let e1 in the definition of convergence of {Tn} to T in L(X,Y). Show that%3D||Tn|| < M, Vn E N, where M = sup{||T||, ||T2||,, |[TN-1||, 1+ ||T||}, for some N E N.

Answered: Image /qna-images/answer/2cb4df61-e158-467b-ab49-eec1a5be15b6.jpg

Answered: For X and Y normed linear spaces, let…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

Suppose that 7 is a Riemann integrable function on [", $]. Prove, in two lines only, that if 7 is continuous on [", $] then 7 is bounded on [", $].
Q6. Let X = (0, 3) equipped with the usual metric d. Зп - 1, Consider the sequence {r„} = {- (a) Is {rn} a convergent sequence? Why? (b) Is {r„} a Cauchy sequence? Justify your answer. (c) Is the metric space ( (0, 3), d ) complete? Prove your answer. (d) Answer the previous parts for X = (0, 3].
The function f : (5,0) → R given by f(x) it is not uniformly continuous on (5, 0). is continuous on 2x-1 (G, 00), prove that
2- VT-I (b) i) Compute lim ii) Hence prove the limit in i) above by e-ố definition. iii) Let a, >0 for all n € N. If a, converges, doesE Vanan+1 converge too? n=1 n=1 2)
29) Identifa of convergen'ce af a power seri's Ž (-3) nx the itervel n-l Ansus: a ) b) c) d) -32x23 e) 3
Let $ g $ be a bounded function on [0,1] and assume that the restriction of $ g $ to the interval [1/n,1] is Riemann integrable for every $ n \geq 2 $. Show that $ g $ is Riemann integrable over the entire interval [0,1]. (Hint: Let $ \epsilon > 0 $ be given and let $ M > 0 $ be a constant such that $ |g(x)| \leq M $ for all $ x $ in [0,1]. Choose $ n \geq 2 $ such that $ \frac{1}{n} < \frac{\epsilon}{3M} $ and note that $$ \sup\{g(x): x \in [0,1/n]\}1/n - \inf\{g(x): x \in [0,1/n]\}1/n < \frac{2\epsilon}{3} $$ Now use that $ g $ is Riemann integrable on [1/n,1] to find an appropriate partition of [0,1].)
4. Let (#1, 1) = (0,0) and let (an+1, Yn+1) = (2 + Yn, 3 – Tn) for n > 1. Prove that (&n, Yn) is a contractive sequence and find its limit.
3. Let {gn} be a sequence of integrable functions such that gn →g a.e. with g integrable. Show that lim f [gn – g| = 0 if and only if lim f \gn| = S \g].
Find the limit using l hopital rule.
Show that if (sn,tn) converges to (s,t) in ℝ2 with the usual d2 metric, then the sequence converges to the same limit in the d0 ("max") metric.
Iff is defined on [a, co ) and integrable on [a,c] for c> a and lim c-8a True O False If f is integrable on [a,b], then F(x) = O True False True O False =[^ƒ« a Let / be a function bounded on [a,b] and let P= {xx,...} be a partition of [a, b]. The lower sum off with respect to P is L(S.P) = Em Ax. Ax=₁-1-1 O True O False f(1) dr= ∞o, then f is improperly integrable on [a, co ). f(t) dt for x = [a, b] is uniformly continuous on [a,b]. Let DC R. A function f:D-R is continuous at c E D iff for any sequence The lower integral of a function f bounded on [a, b] is L(f) = inf{L(f,P): P is a partition of [a,b]), where L(f,P) is the lower sum off with respect to P. O True False O True ⒸFalse where m = inf(f(x) : x (x,-₁,1} and (x₁) in D that converges to c, the sequence e (f(x)) c converges to f(c). Let f:D-R, where DCR, and let c ED'. Then limf(x)=LER iff there exists a sequence (x) in D that converges to c with x #c for all n EN such that the sequence (()) converges to L.

Recommended textbooks for you

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Mathematics For Machine Technology

Advanced Math

ISBN:

9781337798310

Author:

Peterson, John.

Publisher:

Cengage Learning,

Basic Technical Mathematics

Advanced Math

ISBN:

9780134437705

Author:

Washington

Publisher:

PEARSON

Topology

Advanced Math

ISBN:

9780134689517

Author:

Munkres, James R.

Publisher:

Pearson,

SEE MORE TEXTBOOKS