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Exercise 2.8. Let p ï¿½= q; consider the norms of ï¿½p and ï...

Exercise 2.8. Let p ï¿½= q; consider the norms of ï¿½p and ï¿½q on the linear space f. Then these norms are not equivalent. Hint: use the sequence gn as above. pr p∞ Exercise2.11.Provethatifp<r,thenï¿½ ⊂ï¿½. Hint: firstprovethatï¿½ ⊂ï¿½ . Next, given x ∈ ï¿½p, renormalise...

5. Let X be a K-vector space with a norm I ||x Let Y C X b...

5. Let X be a K-vector space with a norm I ||x Let Y C X be a linear subspace. Further let f Y - K be a linear and bounded mapping. (i) Justify why the map p :X - R given by p(x) == is a serni-norm on X. (ii) Using this, together with an appropriate form of the Hahn- Banach Theorem, justify...

1. Denote by Cb,even (R;R) the set of all even, bounded and...

1. Denote by Cb,even (R;R) the set of all even, bounded and contin- uous functions u : R R. Prove that Cb,even(R;R) is a subspace of Cb(R;R), and that the set, {pk: : R R/Px(x)==22, E E Nu {0} } is a linearly independent subset of Cb,even(R;R) which is not a Hamel basis. 2. Using integrat...

1. Let X be a normed space. Let C X be a linearly independe...

1. Let X be a normed space. Let C X be a linearly independent subset and let CF. Prove there is f E X* such that f(ei) = ai, for i = 2. Let {In) be a sequence in F such that < 00 (i.e converges ) for every {yn}ne-1 E n=1 Prove that {In) E Lg, where q is the conjugate to p. Hint: The Pri...

4. Let [a,b] C R be a compact interval and let k: [a,b] X [...

4. Let [a,b] C R be a compact interval and let k: [a,b] X [a,b] R be a continuous function. Define for u € C([a,6];R), b (i) Explain why we can deduce that Kop C((a,b);R) C([a,6];R) and also that Kop is a linear and bounded operator. (It is sufficient here to give the basic ideas with...

7. Let M(n; C) denote the vector space of all complex n x n ...

7. Let M(n; C) denote the vector space of all complex n x n matrices where n Ð„ N. Prove that, < . , . > : M (n; C) x M(n; C) -> C, < A, B > = tr (AB*) defines a scalar product on M(n, C), where tr(C) denotes the trace of the matrix C Ð„ M(n; C).

3. Define T:l2 by, == where No = Nu(0). Prove that wher...

3. Define T:l2 by, == where No = Nu(0). Prove that where |-|2 denotes the norm in I2 and the norm in C 6. Let H be a separable Hilbert space and (ex)REM be anorthonormal basis for H. (i) By considering - for any k,1 E N with k + 1, deduce if the sequence (ex)REN can have a limit in H or n...

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