D_(HK)空间性质的几点注记

Some notes about properties of D_(HK) space

为了对Henstock-Kurzweil可积分布空间(D_(HK)空间)结构有一个更加直观、具体的认识,根据D_(HK)空间的定义、其上的序关系及其范数,证明D_(HK)空间是Banach格;通过证明D_(HK)空间的范数是M-范数,得到D_(HK)空间为AM-空间;利用D_(HK)空间与Bc空间格同构,证明D_(HK)为可分空间;进一步证明Lebesgue可积函数空间在D_(HK)空间所示的序与范数下成Banach格,并且在其上稠密。最后给出D_(HK)空间上的不动点定理,并用此证明椭圆型微分方程解的存在性。

To understand intuitively and concretely the structure of the space of Henstock-Kurzweil inte- grable distributions, DHK, it is shown that DHK is a Banach lattice by the help of the concepts of order and norm. It is also concluded that the DHK space is an AM-space by proving its norm is M-norm. Further- more, DHK is lattice isomorphic to Bc and it is separable. Finally, it is also proven that the space of Lebe- sgue integrable functions is a Banach lattice and it is dense as an ideal in DHK space. Then, a fixed point theorem in DHK space is investigated and is applied to check the existence of solutions to nonlinear sec- ond-order elliptic operators.