Test function spaces $C_p({\Bbb R}^n)$ with $L^p$-norm convergence of the test functions and their dual spaces $C'_p({\Bbb R}^n)$

Abstract

The topology of the test function space $D(R^n)$ is the topology of uniform convergence on compact subsets of $R^n$ of the test functions and their derivatives of all order. In this thesis, instead of the uniform convergence, we consider the convergence of them with respect to $L^p$-norm for each p such that $1 ≤ p ＜ ∞$. We construct the countable norms concerned with each $L^p$-norm. By using those norms, we obtain another test function space $C_p(R^n)$ for each p. We find the necessary and sufficient conditions of the convergence in $C_p(R^n)$ and consider the relation of inclusion between each $C_p(R^n)$. We find alternative characterizations of an element of the topological dual space $C_p``(R^n)$ for each $p$, and consider other properties of the space $C_p``(R^n)$.