关于复数域上Banach空间上的向量值函数的一些注记

Some notes on the vector valued functions in Banach space over complex field

首先证明了取值于Banach空间上强连续的向量值函数是可积分的;然后用初等方法证明了向量值函数的柯西积分公式和高阶导数公式;最后讨论了取值于lp(p≥1)空间上的向量值函数解析、可积、柯西积分公式和高阶导数公式与其各分量函数的关系和表示形式,并且给出了有限维赋范线性空间上的向量值函数连续、解析、可积、柯西定理、柯西积分公式和高阶导数公式与其各分量函数的关系和表示形式.

The strongly continuous vector valued functions in Banach space are proved integrable. The Cauchy integral formula and higher derivative formula of vector-valued functions are proved using the elementary method. The relationship between vector valued functions in the space of l^p （p≥1） and their eomponential functions about analyticity, integrability, the Cauchy integral formula and higher derivative formula are discussed. And the relationship between vector valued functions in finite dimensional normed linear space and their componential functions about continuity, analyticity, integrability, Cauchy＇s theorem, the Cauchy integral formula and higher derivative formula are also given.