On the homogeneous Banach spaces of distributions on \(\mathbb{R}^n\) (Q2737616)

The authors have organized and employed some known concepts and results given by R. E. Edwards (1979, 1982), W. Rudin (1974), A. Wilansky (1978) and K. Yosida (1968) to define and discuss locally convex spaces of distributions on the circle group \(G\). Banach spaces of distributions on \(n\)-dimensional Euclidean space \(\mathbb{R}^n\) is defined to establish some results on these spaces. Banach space of distributions and homogeneous Banach space of distributions are adequately defined, and some interesting results concerning these are evaluated. Concepts of weak topology, closed graph theorem, complex Borel measure, compact support, Bochner integrable function and convolution are used to evaluate results in this paper. For \(\upsilon\) in a homogeneous Banach space distribution \(BD(\mathbb{R}^n)\)-space \(E\) and \(\mu\) in \(M\) (the space of all measures), \(\mu*\upsilon\) is defined as a distribution, even when neither \(\mu\) or \(\upsilon\) has compact support and it is shown that \(\mu*\upsilon\) is the limit in \(E\) of finite linear combinations of translates of \(\upsilon\). It is also established that, if \(E\) denotes any one of \(L^p\) \((1\leq p<\infty)\) or \(C^\infty\) \((0\leq m\leq \infty)\), then \(E= L^1*E\), and further this is generalized by taking \(E\) to be any homogeneous \(BD(\mathbb{R}^n)\)-space by employing Hewitt's formulation [cf. R. E. Edwards (1982)].NEWLINENEWLINEFor the entire collection see [Zbl 0960.00033].