12. Suppose that a, b R and a < b. Show that the vector space C[a, b] of all continuous complexvalued functions defined on [a, b], with supremum norm||f||is a Banach space.sup f(t)\,te [a,b]

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Answered: 12. Suppose that a, b E R and a < b.…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Let V be the set of sequences of real numbers. For {an},{bn}belongs to V and any real number t, define {an} +{bn} = {an+ bn} and t{an} = {tan}. Prove that, with these operations, V is a vector space over R.
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Q 06: For n≥0 , the set Pn of polynomials of degree at most n consists of all polynomials of the form p(t) = a0 + a1t + a2t2 +a3t3+......+ antn form a vector space. Where the coefficients a0, a1, a2, ..... an and the variable t are real numbers.
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Let V be the vector space of polynomials in over R of degree ≤ 3 and define an inner product f . g = f¹ f(x)g(x)dx for all f, g € V. You may assume that this makes V into an inner product space. Define the linear mapa: V → V by a(ƒ)(x) = f(x) + xƒ'(x) where the f' denotes differentiation. Which of the following holds with respect to this inner product? Select one: O O O a is self-adjoint a is orthogonal a is 'anti-self adjoint' meaning a* = -α a = I-a, where I denotes the identity map None of the others apply

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12. Suppose that a, b E R and a < b. Show that the vector space C[a, b] of all continuous complex valued functions defined on [a, b], with supremum norm is a Banach space. ||f||= sup f(t)\, te [a,b]