Source Notre Dame J. Formal Logic , Volume 50, Number 4 , - Zentralblatt MATH identifier Keywords computable analysis reverse mathematics weak Konig's lemma Hahn-Banach extension theorem multivalued functions. Gherardi , Guido; Marcone , Alberto.

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Notre Dame J. Formal Logic 50 , no. Thus T is subadditive and consequently T is sublinear on E. It is easy to check that T is positively homogeneous, so to prove that T is sublin- ear it remains to prove that T is subadditive. Define T as in Lemma 3. In the following we prove Theorem 2. The proof follows directly from Theorem 3. The following theorem is a generalization form of the classical Hahn-Banach the- orem for sublinear and superlinear functionals.

Obviously, the necessary condition follows directly from the above state- ment. Then from Corollary 3. The proof follows directly from the definition of a sublinear functional S, Corollary 2. From the inequality 3. The following example shows that the converse of the above lemma does not al- ways true. Then we show that P is a superlinear functional on E, S is a sublinear functional on E, f0 is a linear functional on E, which satisfies the inequalities 3.

It easy to verify that f0 is a linear functional on E, P is a superlinear functional on p E and S is a sublinear functional on E. Finally, suppose that the inequality 3. Then f0 cannot satisfy the inequality 3. The sufficient condition is satisfied from Lemma 3. Conversely, let con- ditions 3. Now it easy to check that T is positively homogeneous, so to prove that T is sublinear it remains to show that T is subadditive.

## Gherardi , Marcone : How Incomputable Is the Separable Hahn-Banach Theorem?

The necessary condition follows directly from Lemma 3. Moreover, from Lemma 3. This means that theorem 3. The proof follows directly from Corollary 2. Then F is a maximal for A.

So L must be defined on E, otherwise we can extended it contracting the fact that it is maximal, the theorem follows. Acknowledgment The authors would like to express thier deepest gratitude and thanks to full Doc- tor, Prof. Part of the work for this paper was done while the second author was visiting the Faculty of Mathematics and information technology, Belgorod States University, Belgorod, Russia and Also, department of Information Systems and Technologies, Vologda State Technical University.

The author would like to thank the Faculty of Mathe- matics and information technology , Belgorod States University, Belgorod,and also the department of Information Systems and Technologies Vologda State Technical University, Vologda for its hospitality. References 1. Kelley, I. Hasumi, M. Tohoku Math.

## On fuzzy order relations

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