1* Let X be the space of sequences of real numbers with only finitely many nonzero terms, withthe norm ||x|| = sup en il, for every x = (§;). Let T : X → X be defined byjEN1Tx = (61.:11for every x = (E;) = (§1,§2,. ).(a) Show that T is linear and bounded, and find ||T||.(b) Show that T : X → X is bijective.(c) Show that the inverse T-l of T is not bounded.

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Answered: 1* Let X be the space of sequences of…

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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24. Let f(x) be a convex function on R¹. Let a be a fixed vector on R" and let x be a fixed real number. Define g(x) on R" by g(x) = f(a x + x). Show g(x) is convex on R" but that if n ≥ 2, then g(x) is not strictly convex. Deduce that g(x, y, z) = (4x + 5y - 8z + 17)8 is convex but not strictly convex on R³. State why this does not follow from Theorem (2.3.10) (c).
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Let S be a non-empty set of real numbers which is bounded above, and suppose that x = sup(S). (a) Prove that for all € > 0 there exists s ES so that s = (x − €, x]. (b) Suppose now that x S. Prove that for every e > 0 the set {s ES | se (x - e, x]} is infinite.
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Suppose f: (0, ∞) → (0, ∞) and g: (0, ∞) → (0, ∞) and h: (0, ∞) → (0, ∞) are continuous functions such that h(x) = f(x)g(x) for every positive real number . True or false: If f and h are differentiable, then g must be differentiable too. True False
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Suppose f: (0, ∞) → (0, ∞) is a continuous function, and g: (0, ∞) → (0,00) is defined by the property that g(x) = (f(x))² for every positive real number . True or false: If g is differentiable, then f must be differentiable too. True False

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