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3.1. Let X be a vector space and d a metric on X. Show that a pair (X, d) is a metric vector space if for sequences (x,) and (yn) of vectors in X and sequences (An) of scalars in F, these conditions hold: (a) Xn - x, Yn -y imply (xn + Yn) (x +y), (b) - x imply Anan - Ar.

3.1. Let X be a vector space and d a metric on X. Show that a pair (X, d) is a metric vector space if for sequences (x,) and (yn) of vectors in X and sequences (An) of scalars in F, these conditions hold: (a) Xn - x, Yn -y imply (xn + Yn) (x +y), (b) - x imply Anan - Ar.

Advanced Engineering Mathematics
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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Exercises 3.1. Let X be a vector space and d a metric on X. Show that a pair (X, d) is a metric vector space if for sequences (z,) and (yn) of vectors in X and sequences (An) of scalars in F, these conditions hold: (a) Xn - x, Yn → y imply (xn + Yn) - (x + y), (b) An → A, xn - imply Anxn Ax. 3.2. For a norm || || on a vector space X, show that 11 (a) ||0|| = 0, (b) ||x||>0 if x #0 (cf. Exercise 2.7). 3.3. Let d be the discrete metric on a nontrivial vector space X. Show that (X, d) is not a metric vector space. 3.4. Show that the sum of two seminorms is a seminorm. 3.5. Let B = {x;}ieJ be a Hamel basis in a vector space X. Show that |||| = |A1|+ +Anl, for x = A1ri +.. + An rin, is a norm on X. 3.6. Show that the seminorm p on a seminormed space (X, p) is a continuous function. Also show that the norm is a continuous function on a normed space. 3.7. Prove the triangle inequality for the normed space . 3.8. Show that for x = (r1,... , Tn) E F", |||| lim ||a||p- 3.9. Complete the proof of Theorem 3.3.
Transcribed Image Text:Exercises 3.1. Let X be a vector space and d a metric on X. Show that a pair (X, d) is a metric vector space if for sequences (z,) and (yn) of vectors in X and sequences (An) of scalars in F, these conditions hold: (a) Xn - x, Yn → y imply (xn + Yn) - (x + y), (b) An → A, xn - imply Anxn Ax. 3.2. For a norm || || on a vector space X, show that 11 (a) ||0|| = 0, (b) ||x||>0 if x #0 (cf. Exercise 2.7). 3.3. Let d be the discrete metric on a nontrivial vector space X. Show that (X, d) is not a metric vector space. 3.4. Show that the sum of two seminorms is a seminorm. 3.5. Let B = {x;}ieJ be a Hamel basis in a vector space X. Show that |||| = |A1|+ +Anl, for x = A1ri +.. + An rin, is a norm on X. 3.6. Show that the seminorm p on a seminormed space (X, p) is a continuous function. Also show that the norm is a continuous function on a normed space. 3.7. Prove the triangle inequality for the normed space . 3.8. Show that for x = (r1,... , Tn) E F", |||| lim ||a||p- 3.9. Complete the proof of Theorem 3.3.
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