s.n.函數(Phi)(pi)(x_{1},...) (在[8]內有詳細討論)定義出B(pi)空間,並且證明出B(pi)是B(Pi)^0 (在B(Pi) (用s.n.函數(Phi)(Pi)(x_{1},...)定義出的可解析函數空間))內多項式的閉包)的對偶空間,這些證明須在{pi_{n}}滿族一些正則條件.在這篇論文中,我們將證明事實上我們有以下關係:(B(Pi)^0)^*幾乎等於B(pi),B(pi)^*幾乎等於B(Pi).這是有趣將古典算子理想(S(Pi))^(0),S(pi)和S(Pi),或用(Phi)(Pi)和(Phi)(pi)所定義出的s.n.理想之對偶性做類比.

The space B(pi), defined by s.n. function (Phi)(pi)(x_{1},...), were discussed in details in [8], and it is shown that B(pi) is the dual space of B(Pi)^0, which is the closure of the polynomials in B(Pi), the space of analytic function defined by the s.n. function (Phi)(Pi)(x_{1},...), provided that {pi_{n}} satisfies some regularity condition. In this article, we will show that in fact we have the relation (B(Pi)^0)^* approx B(pi), B(pi)^* approx B(Pi). This is an interesting analogy to the classical duality between the operator ideal (S(Pi))^(0), S(pi) and S(Pi), or, the s.n. ideal defined by (Phi)(Pi) and (Phi)(pi).

Chapter1: Introduction................................1
Chapter2: The space B(Phi) and its properties................................4
2.1 Space of analytic function defined by s.n. function................................4
2.2 The s.n. function (Phi)(Pi) and (Phi)(L)................................6
2.3 The space B(Pi)^0, B(pi) and B(Pi)................................8
2.4 The dual of B(pi)................................15
References...............................................................21

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