Volume 12, 2000

University of Bialystok

Copyright (c) 2000 Association of Mizar Users

**Noboru Endou**- Shinshu University, Nagano
**Katsumi Wasaki**- Shinshu University, Nagano
**Yasunari Shidama**- Shinshu University, Nagano

- This article is an introduction to convex analysis. In the beginning, we have defined the concept of strictly convexity and proved some basic properties between convexity and strictly convexity. Moreover, we have defined concepts of other convexity and semicontinuity, and proved their basic properties.

- Some Useful Properties of $n$-Tuples on ${\Bbb R}$
- Convex and Strictly Convex Functions
- Definitions of Several Convexity and Semicontinuity Concepts

- [1]
Grzegorz Bancerek.
The ordinal numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [2]
Grzegorz Bancerek and Krzysztof Hryniewiecki.
Segments of natural numbers and finite sequences.
*Journal of Formalized Mathematics*, 1, 1989. - [3]
Jozef Bialas.
Infimum and supremum of the set of real numbers. Measure theory.
*Journal of Formalized Mathematics*, 2, 1990. - [4]
Jozef Bialas.
Properties of the intervals of real numbers.
*Journal of Formalized Mathematics*, 5, 1993. - [5]
Czeslaw Bylinski.
Functions and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [6]
Czeslaw Bylinski.
Partial functions.
*Journal of Formalized Mathematics*, 1, 1989. - [7]
Czeslaw Bylinski.
Binary operations applied to finite sequences.
*Journal of Formalized Mathematics*, 2, 1990. - [8]
Czeslaw Bylinski.
Finite sequences and tuples of elements of a non-empty sets.
*Journal of Formalized Mathematics*, 2, 1990. - [9]
Czeslaw Bylinski.
The sum and product of finite sequences of real numbers.
*Journal of Formalized Mathematics*, 2, 1990. - [10]
Krzysztof Hryniewiecki.
Basic properties of real numbers.
*Journal of Formalized Mathematics*, 1, 1989. - [11]
Jaroslaw Kotowicz.
Real sequences and basic operations on them.
*Journal of Formalized Mathematics*, 1, 1989. - [12]
Jaroslaw Kotowicz and Yuji Sakai.
Properties of partial functions from a domain to the set of real numbers.
*Journal of Formalized Mathematics*, 5, 1993. - [13]
Jan Popiolek.
Some properties of functions modul and signum.
*Journal of Formalized Mathematics*, 1, 1989. - [14]
Konrad Raczkowski and Pawel Sadowski.
Real function continuity.
*Journal of Formalized Mathematics*, 2, 1990. - [15]
Konrad Raczkowski and Pawel Sadowski.
Topological properties of subsets in real numbers.
*Journal of Formalized Mathematics*, 2, 1990. - [16]
Andrzej Trybulec.
Tarski Grothendieck set theory.
*Journal of Formalized Mathematics*, Axiomatics, 1989. - [17]
Andrzej Trybulec and Czeslaw Bylinski.
Some properties of real numbers operations: min, max, square, and square root.
*Journal of Formalized Mathematics*, 1, 1989. - [18]
Zinaida Trybulec.
Properties of subsets.
*Journal of Formalized Mathematics*, 1, 1989. - [19]
Edmund Woronowicz.
Relations and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989.

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