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Then ~l and ){2 equable ==-dp ~l ~/ ~2 equable. Proof. From the set-theoretic equality 18 ,(X l v X2) = I~ X1 u I & . (X I ~ ~2) =~v. ~lV~V. ~2" From this, the lemma follows at once. 6. lg - Equable pseudo-topological vector spaces. e. 2), that each topological vector space is equable. However, not all pseudotopological vector spaces are equable (*). Given any pseudo-topological vector space E, we can introduce on ~ a new pseudo-topology, thus obtaining a new pseudo-topological vector space E I~" .

1) Let ~ E 1 ~ o Hence ~ ~ = ~ E 1 , t(~)~(~) =~ (~V~) = W - [ ( ~ ) ~E 2, ~E2~. , Proposition. 10)). Proof. lO) it is sufficient to show that b: El~ x E 2 ~ E 3 ~ • i~Ei ~ , i = 1,2. is continuous at the point (0~0). So let Hence ~i ~ ~ i = ~ V ~ i ~ E i , and b(~l,~2)~-b(~l,~2 ) = b(W~l,~2) shows that b(~l, @2 ) ~ E 3 ~ " =~V b ( N l , ~ 2 ) ~E 3, which - § 3. 32 - DIFFERENTIABILITY AND DERIVATIVES. In this section, the definition of differentiability is given and the most elementary results of calculus are proved.

X ( U and - x ( U o is convex , then z = ~x, where Since ~ O , ~ . U = U, ~x = I ~ l (~x) @ U. Thus we have V ~ I l ~ / , which shows that V e ~ V ~ o e) Let V 6 1 ~ , also ½U e ~ ; and choose U as before. By (c) we have and since ½U is also convex it follows that ½U ~ . 2)). 7) are necessary and sufficient in order that ~ i s the neighborhood-filter of zero for a unique compatible topology on ~ (cf. [Q] ). 9) Proposition. 24 - For any pseudo-topological vector space E, the space E° defined above is a locally convex topological vector space.