Lemma 2.31 Let (X, || - ||x), (Y, || - ||Y) be normed vector spaces and F E L(X,Y).Then||F||c(x,Y) :=|| Fæ||y =||Fx||y.suprɛX; ||r||x=1suprɛX; ||r||x<1Proof:excersises.

Answered: Image /qna-images/answer/18d32594-fa68-45f9-8d5f-ad6538ff76b1.jpg

Answered: Lemma 2.31 Let (X, || · ||x), (Y, || ·…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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2: Let a function d: R2 x R? →R defined by d(x, y) = ((x1 – y1)² + (x2 - y2)?) Vx = (x1, x2), y = (y1, Y2) E R2, then d be a %3D metric function on R2.
Lemma 2.31 Let (X, || · ||x), (Y, || · ||Y) be normed vector spaces and F E L(X,Y). Then ||Fcx,Y) sup xƐX; ||x||x=1 ||Fx||Y sup xƐX; ||x||x<1 ||Fx||y. Proof: excersises.
7. If r = (x, y, z) and F vector r) is there a value p so that div F = 0? HE (where r| denotes the length of the
Definition 28. Let V be a vector space over F (here F= R, C). Then the function (•₁•): V × V → F is said to be an inner product on V if the following conditions are satisfied: (x, x) is real and (x,x) ≥ 0 for all (x,x) = 0 if and only if x € 0₂. (i) (ii) (iii) (x, ay) = a(x, y) for all x, y V (iv) (x, y+z) = (x, y) + (x, z) for all (v) (x, y) = (y,x) for all x, y V. (a) (.,.): R¹ × R¹ → R is defined as xεν. (x, y) = x¹y, and for all x,y,z € V. Example: Verify that each of the following mappings define a real inner product. a EF. for all x, y ER".
1. Let f : R" → R" be a function with the property that for any v, w E R" it holds that f(v) · f(w) = V· W. I.e. f preserves the inner product on the vector space R". Show that f is a linear transfor- mation.
44. Products of scalar and vector functions Suppose that the sca- lar function u(t) and the vector function r(t) are both defined for a

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Lemma 2.31 Let (X, || · ||x), (Y, || · ||Y) be normed vector spaces and F E L(X,Y). Then ||F||c(x,Y) := sup rɛX; ||a||x=1 || Fæ||y = sup rƐX; ||r||x<1 || Fe||y.