Image Under a Linear Mapping

This chapter presents some properties of the image under a linear mapping, and estimates the norms in the Banach spaces. It characterizes continuous multilinear mappings. The chapter states that T be a uniformly continuous mapping from U into a Neumann space F, where U is sequentially dense in a subset X of a separated semi-normed space E. It provides the continuity of the image under T , and Holder's general inequality. Continuity follows from the norm inequality, according to the characterization of continuous multilinear mapping. The chapter also shows that the value of a locally -integrable measure on a test function equals the integral of their product.