标题： 标量积的初等分布 显示英文标题

标题： Scalar products of elementary distributions

摘要： 实数域被扩展为一个更大的交换域，我们研究了为有限离散支撑的分布定义标量积的可能性。 然后我们关注最简单的可能扩展（这是一个有序域），我们为此标量积提供了显式公式，并展示了一个正定内积空间的结构。 在一维应用中，针对薛定谔方程，由原点支撑的分布被嵌入到一个括号-矢量空间中，在其中“奇异”势能以自然的方式定义，用于描述点相互作用。 可以与非标准分析中出现的超实数建立联系，但这不是必要的，我们对$\bf R$和$\bf C$的扩展是通过一种相当基础的方法获得的。 显示英文摘要

摘要： The field of real numbers being extended as a larger commutative field, we investigate the possibility of defining a scalar product for the distributions of finite discrete support. Then we focus on the most simple possible extension (which is an ordered field), we provide explicit formulas for this scalar product, and we exhibit a structure of positive definite inner-product space. In a one-dimensional application to the Schroedinger equation, the distributions supported by the origin are embedded into a bra-ket vector space, where the "singular" potential describing point interaction is defined in a natural way. A contact with the hyperreal numbers that arise in nonstandard analysis is possible but not essential, our extensions of $\bf R$ and $\bf C$ being obtained by a quite elementary method.