By G. Coeuré (auth.), Leopoldo Nachbin (eds.)

**Read Online or Download Functional Analysis and Applications: Proceedings of the Symposium of Analysis Universidade Federal de Pernambuco Recife, Pernambuco, Brasil, July 9–29, 1972 PDF**

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**Additional resources for Functional Analysis and Applications: Proceedings of the Symposium of Analysis Universidade Federal de Pernambuco Recife, Pernambuco, Brasil, July 9–29, 1972**

**Example text**

Y d 0 4 e. d F E(LF). o n4 : R VEU(F). From (I,j) we (V nF k ) <;;: W, and the lemma is proved. 4 well-locat e d; -6Ub4pace. ~ghbMhood R belong~ to U(R). Then (i) ~ (ii) ~ (iii) (iv) ~ ~ (v) ~ (vi) . lent. PROOF: The implication lR T (R, R~). Hence (ii) (i) ~ ~ (iii). by means of the Hahn-Banach theorem. (2,9». e. <====? (i) F (ii) is obvious. Since Rl E: (LF), (iv) is a reformulation of (v) ~ (vi) is (i) trivial (cf. is a Schwartz space, then by (1,24), the in (1) is a homomorphism iff I is a weak homomorphism, (v) .

Holds for all absolutely convex E' ,. , if closed subspace N of E/N is complete for an arbitrary E. If (c) denotes the class of all complete spaces, then the rela- tionships between the above types of completeness can be summarized as follows: (1,6) "THEOREM": ? smulian for the case of Banach spaces in [54J by (cf. M. also [36J). ptak [69, 61J who studied them in great detail. (B r ). ) was known for a long time that 2) ptak ? (c) (he~). com- This is interesting, for it ~ (he~) (cf. KMhe [50J).

As the X II (X) set of m I all formal linear combinations (lJiE: lK, Xi E: X). e. A(V): II (X) ---- F. x. e. 70* Ox (X,t) TI sup ) E: T 2 . Then (A(X), lOX) t gives -1 t"d wx '7' ("'X). Therefore, follows. It remains to show that F = (F ,'0) The continuity of 7;/E T l , which gives lished. A(X) '7' (1). With = w-1 ( (Q). implies that = 0 (II (X), and thus also (2) has the property and T I · set X f E: C (X) } • : * Wx-1 1'0) = 0*<;;: oX. ) WH , E = sup f€H 61.. ned a,6 the. e.. P. Every (3,7), Furthermore, VO p To (it) OC 0 de) E T I , hence E T2 .