凸泛函和导算子的特性及凸性问题

Specificity of Convex Functional and Problems of Convexity in LCS

本文引入了导算子的正定及广义正定的概念,研究了凸泛函的各种性质,并讨论了凸泛函与它的导算子之间的关系及泛函存在极值的一些条件。最后讨论了空间的一些凸性问题。 §1 凸泛函和导算子的特性 定义1.1:设D是线性空间E中的一个凸集,f(x)是D上的一个实值函数,如果 f[λx+(1-λ)y]≤λf(x)+(1-λ)f(y)对λ∈(0,1)和x,y∈D成立,则称f(x)是D上的凸泛函。

In this paper I give several equivalent propositions about convex functional and its derived operator in a linear normed space, introduce the conception of positive definite and generalized definite of derived operator, and obtain some meaningful properties of convex functional. Meanwhile I study the relation between convex functional and its derived operator,give some condition of functional existing extreme value. At last I deal with the problems of convexity in LCS, extend all kinds of convexity in Banach space to the locally convex space.