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HW7_2011 copy

# HW7_2011 copy - b Show that I is a bounded operator that is...

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Math 602 Homework 7 Please note the following very useful inequalities: | f ( t ) | ≤ k f k for all t S if k f k = sup x S | f ( x ) | and Z b a f ( t ) dt Z b a f ( t ) dt for a < b 1. Consider the sequence of functions f n ( x ) = x n . a) Show that the sequence f n is point-wise convergent for all x [0 , 1], that is, show that for all x [0 , 1] the limit lim n →∞ f n ( x ) = f ( x ) exists; what is the limit function f ( x )? b) The functions f n belong to the space C [0 , 1] of continuous functions on [0 , 1]. Do they converge in the sup norm in this space? c) The functions f n belong to the space L 2 [0 , 1]. Do they converge in the L 2 norm in this space? 2. Consider the integral operator I : C [0 , 2] C [0 , 2] given by If ( x ) = Z x 0 f ( t ) dt a) Show that I is a linear transformation.
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Unformatted text preview: b) Show that I is a bounded operator, that is, there exists a number B so that k If k ∞ ≤ B k f k ∞ for all f ∈ C [0 , 2] and ﬁnd the (smallest) number B . 3. Consider the diﬀerentiation operator D : P → P (acting on polyno-mials): Df = df dt a) Show that D is a linear transformation. b) Equip P with the sup norm on [0 , 2]. Show that D is an unbounded operator: consider the polynomials f n ( t ) = t n and show that there is no constant B so that k Df n k ∞ ≤ B k f n k ∞ for all n . 1...
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