article · 1972

VECTOR VALUED KOTHE FUNCTION SPACES III BY AIN L.

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Abstract

This paper is a continuation of [8] and [9]. We begin with a general situation. The results obtained will yield information about compact sets in v.f.s.’s. Let X and a family {Y a e A} be Hausdorff topological spaces and let X--. Y be continuous maps. We suppose that ($) (x) for each a implies z x. Define Y and:X- * Y by (x) {(x)}. Then is continuous and one to one. A set S _ is said to be projec$ivvly cornpac if (S) is compact for each a. A sequence (x) X is projectively convergent if a(z) is convergent for each a. Other terms are defined similarly. The proofs of the following propositions present no difficulties; the proof of Proposition 1.1 uses Tychonoff’s theorem and that of Proposition 1.3(2) uses the finite intersection property characterization of compactness (see [6]). PROPOSITION 1.1 A se $ S X is compact if and only if (1) S is projectively compact, and (2) every projectively convergent net in S is convergent to a point in S. PROPOSXTO 1.2. Suppose A is countable. Then a set S X is sequentially compact (respectively relatively sequentially compact) if and only if (1) S is projec$ively sequentially compact (respectively relatively sequentially compact), and (2) every projec$ively convergen sequence in S is convergent to a point in S (respectively convergent). PROPOSTO 1.3. Suppose A; is countable. Then&quot; (1) If in Y, the compact (respectively relatively compact) sets are sequentially compact (respectively relatively sequentially compact), then he same is $rue in X. (2) If in Y the sequentially compact sets are compact, then the same is true in X. (3) If inY,, the countably compact sets are compact, then the same is true in X. (4) If in Y the countably compact (respectively relatively countably conpact) sets are sequentially compact (respectively relatively sequentially compact), then the same is true in X. We now make some applications of the above three propositions. THEORE 1.4. If E is rnerizable and S(E) is a v.f.s, with a topology finer than the weak topology induced from (E) then the compact, sequentially compact,

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APA 7

(1972). VECTOR VALUED KOTHE FUNCTION SPACES III BY AIN L.

MLA 9

“VECTOR VALUED KOTHE FUNCTION SPACES III BY AIN L.” 1972.

Chicago

1972. “VECTOR VALUED KOTHE FUNCTION SPACES III BY AIN L.”

Harvard

(1972) ‘VECTOR VALUED KOTHE FUNCTION SPACES III BY AIN L’.

BibTeX

@article{anon1972vector,
  title = {{VECTOR VALUED KOTHE FUNCTION SPACES III BY AIN L.}},
  year = {1972}
}

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