By Gian-Carlo Rota

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5. Proof. £ is a Montel space, hence the strong dual E f is also Montel, in particular, barreled. Thus E is distinguished. If K is a compact subset of U with nonempty interior, then /-► su p |/(x )| xeK defines a continuous norm on E. D. 5. Completeness of The question of completeness of (< ? f(l0 ,O was raised first by Nachbin [32]. Partial affirmative answers were given by Dineen, Aron, and Chae. Dineen first proved the completeness of (J f (£), xm) when £ is a Banach space [11], and later he proved the completeness of (J^{E), Tm) when £ is a countable product of Banach spaces [13].

O f Kentucky 7973, p p . 196-212. S p rin g e r-V e rla g , B erlin a n d N e w Y o r k , 1974. 42. M . Schottenloher , D a s L e v ip ro b le m in u n e n d lic h d im e n s io n a le n R a u m e n m it S c h a u d e r z e rle g u n g , H a b ilita tio n T h e s is , L u d w ig M a x im ilia n s U n iv . M ü n c h e n , 1974. 43. H . W h i t n e y , A n a ly tic e x te n s io n s o f d iffe re n tia b le fu n c tio n s d e fin e d in c lo s e d se ts, Trans. Amer. Math. Soc. 3 6 (1934), 63-89.

Let x e E with a(x) < 1. , 7ta(x) = na(y) for some y e Bx. Hence a(x —y) = 0 and therefore P(x) = P(y) for all P e 0 >(mEx). Therefore for all P e ^ ( mEa), we have l|p|U=a(jc) sup \p (x)\ < sup \p (y)\ = <1 yeBa and ■ < ? (mEa) has the induced topology of ¿P(mE). 2. D. For an arbitrary locally convex space E, condition (A ) implies condition (C). 2. 3. For an arbitrary locally convex space E, condition (N ) implies condition (C). Proof. For a < the mapping naP:E/P -►E/a is continuous and surjec tive, and £//?