Ex3:Let X be a normed linear space.Use the Hahn-Banach theorem to prove the following statements.For any non zero x in X, there is a bounded linear functionalO in X' such that ll|| = 1 and o(x)= ||x||-i)%3Dii) If x, y in X verify p(x) = 4p(y) ▼ p in x' , then x = y.

By the application of Hahn-Banach theorem, we have to prove the following statements : i) For any…

Answered: Ex3: Let X be a normed linear space.…

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

Let X be a normed linear space and let P, Q E B(X). Show that thetransformation T: B(X) B(X) defined by T(R) = PRQlinearis bounded.or defined harThe problem is Functional analysis
Suppose f is the function defined by f(x) = Ax, where a. The domain of f is b. The codomain of f is 24 0 DA. OB. OC. OD. DE. OF. A = 0 1 8 3 -3 0 2 -1 5 For instance, enter R^5 for R5. For instance, enter R^5 for R5. c. Select all of the vectors that are in the kernel of f. You should be able to justify your answers (there may be more than one correct answer DA. OB. OC. D. DE. F. d. Select all of the vectors that are in the image of f. You should be able to justify your answers (there may be more than one correct answer).
(b) {T } is a sequence of bounded linear maps on a Banach space X and Tx = lim 1x exists for all xE X. Show that T is a bounded linear operator.
4) Let V be the vector space of polynomials of degree <2 and let p(x) = ax² + bx + c. Find the adjoint operator (T*@)(p) for T:V → V and o:V → R given by d) Tp(x) = x²p(x) + x³p'(x), @(p) = p"(1).
f: X→ Y and g: Y→V be bijective functions. Show the following: ) gof: X → V is injective. ) gof: X→ V is surjective. ) (gof)-¹ = f¹og-¹ v injective if x1 #xn → v(x1) = v(xn)
5. Let W be the space of polynomials p(x) with real coefficients of degree < 10, satisfying p(0) = p(1) = p'(1) = 0. The dimension of W is (a) 1 (b) 7 (c) 8 (d) 9 (е) 11
Let T: R³ R2 be a linear mapping. → 10 *([:]) = [ ] ¹ ({]) - [ ] -~-~(:)) - [²7] ([]) - T = and T find T (ED). Given that T sin (a) Ər f α Ω E
Prove that the set V = {(x, y) : x, y e R} of all ordered pairs with the operations (x, y) © (x', y') = (x + a' + 1, y + y' + 1) for (x, y), (a', y') e V and s0 (r, y) (s +sx-1, s + sy – 1) for s e R is a vector space. Then write (4,0) e V as a linear combination of (1,0) and (0, 1).
Find the kernel and range of each of the following linear operators on P^3(the vector space of polynomials with degree less than 3):(a) L(p(x)) = xp′(x), where p′(x) is the derivative of p(x). Hint: Start with p(x) = ax2 + bx + c. (b) L(p(x)) = p(x) − p′(x).
Let φ1 = 1, φ2 = x, φ3 = x2 on the interval 0 ≤x ≤1. Find the following inner products:(a) < φ1,φ2 >(b) < φ2,φ3 >(c) Distance between φ1 and φ2

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

Ex3: Let X be a normed linear space. Use the Hahn-Banach theorem to prove the following statements. i) For any non zero x in X, there is a bounded linear functional in X such that ||||= 1 and (x) = ||x||. ii) If x, y in X verify 4(x) = (y) in X', then x = y.