By William H Ruckle

ISBN-10: 0273085077

ISBN-13: 9780273085072

Those are lecture notes for a direction entitled "Sequence areas" which the writer gave on the college of Frankfurt through the educational yr 1975-76.

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**Example text**

1. of S Finally, (u(i)1) is Thus, S-weakly bounded subsets are precompact so that T(S,S a ) = a ). 1/I Exercises 1. Let (u) be a sequence having disjoint support and let S be a Köthe space such that S(Un) = m. 50 Given s in S such that Ts # is the smallest index if define r5(n) to be 0. Show that the mapping rs(n)un is a continuous projection from S onto = where S is given its strong topology. Let S be a Köthe sequence space of the form 2. where each = is a Kbthe space which is an FK-space in the strong topology and c S that if the Sa_Mackey topology on a Köthe space S coincides with the normal topology then the Sa_strong topology also coincides with the normal topology.

C c T. 5a Next, Ta C S since T C and S is l1-perfect. ,u —1 and v ,... 5arn) ÷ ])] = =S. 1) /1/ Exercises 1. Show that if S is a nuclear FK-Köthe space with the normal topology the following are equivalent: (a) 42 SCm; 0 (b) S Cl1; (c) S C 2. Prove: 3. Let i0 = (a) for some permutation ii. 10. p>0 (10)a Show that 1110 = 10 and that = m so that the 1°-normal topology on m is nuclear. (b) Show that if S is any normal sequence space such that S a 0. = m then the linear span T of 1 S is normal 11T = T and Ta = m so the T-normal topology on m is nuclear.

Prove: 3. Let i0 = (a) for some permutation ii. 10. p>0 (10)a Show that 1110 = 10 and that = m so that the 1°-normal topology on m is nuclear. (b) Show that if S is any normal sequence space such that S a 0. = m then the linear span T of 1 S is normal 11T = T and Ta = m so the T-normal topology on m is nuclear. (c) Show that if S cm and S has a nuclear normal topology then S = m. (d) Show that fl{T: T = is nuclear is equal to 4. (a) For any set A of indices let of indices in A < n}. {number = Suppose (bn) is any sequence of positive numbers which decreases monotonically to 0 and n a n is any series such that a n n = Show there is a set of indices A such that A' a (b) = °° and limn dn (A) /bn = 0.

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