By Choudhary P.
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Additional info for A practical approach to linear algebra
S) implies that satisfies condition (ii) of Grothendieck's Theorem B4 (in Appendix) and, by the latter Theorem, Thus (i) implies (iii). € ~e A is relatively weakly compact. Now suppose that S is locally compact for the remainder of this proof. We assume that (iii) holds and prove (i). Let C be a given compact subset of S and V be a compact neighbourhood of C. fl. Thus~ E Me(S). Finally we show that (ii) implies (i). Let£> 0 be given. We can choose a compact subset F such that II 1~1-1~1IF II < £.
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First we prove items (i) and (ii). 5. Let u1 ,u2 ,v be open subsets of S such that u1 £ u1 • u 2 £ u 2 • 1 £ V and vu1u 2 S U. l(a) we have that and X £ This completes our proof for (i) and (ii). Second we prove item (iii). Suppose t £ x(O n s 1 )-l and choose w £ 0 n s 1 with tw .. x. · By (ii) and the fact that s 1 is dense we can find v w(W n s1) £ -1 n v n s1 • W n s 1 with vy • w. -1 -1 -1 We note that (tv)V S x(O n s1) for if p E (tv)V then So we can find y £ x = tw • tvy and so p t £ £ £ pVy £ pV(W n s 1) £ p(O n S1) x(O n s 1) -1 • Hence, with item (ii) in mind, we have that int((tv)V-1) S int(x(O n s 1>-1> and so (iii) follows.