Question 2: Let V be an F vector space, || || and || ||' two norms...
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Question 2: Let V be an F vector space, || || and || ||' two norms on V. For each & > 0 and a V. Let B(x, e) and B'(x, e) denotes the e-open ball at æ in (V, ||-|| and (V,||||') respectively. Show that a) ||-|| and || ||' are equivalent →B'(0, 1) C B(0,r) and B(0, 1) C B'(0, R) for some r, R>0 b) Use a) to show that Equivalent norms generate the same topology on V. c) |||| and || ||' are equivalent the identity map I: (V₂||·||) → (V₂ ||·||') is a Homeomorphism. [10,6,10] Question 3: Let a R-{0}. Show that the space C₂[-a, a] of all real-valued continuous functions on [-a, a], with the norm ||²||2 = ( ["* x²(t)dt)}¹/² is not a Banach space. [10] Question 2: Let V be an F vector space, || || and || ||' two norms on V. For each & > 0 and a V. Let B(x, e) and B'(x, e) denotes the e-open ball at æ in (V, ||-|| and (V,||||') respectively. Show that a) ||-|| and || ||' are equivalent →B'(0, 1) C B(0,r) and B(0, 1) C B'(0, R) for some r, R>0 b) Use a) to show that Equivalent norms generate the same topology on V. c) |||| and || ||' are equivalent the identity map I: (V₂||·||) → (V₂ ||·||') is a Homeomorphism. [10,6,10] Question 3: Let a R-{0}. Show that the space C₂[-a, a] of all real-valued continuous functions on [-a, a], with the norm ||²||2 = ( ["* x²(t)dt)}¹/² is not a Banach space. [10]
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Related Book For
Differential Equations and Linear Algebra
ISBN: 978-0131860612
2nd edition
Authors: Jerry Farlow, James E. Hall, Jean Marie McDill, Beverly H. West
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