Abstract
AbstractIn this chapter we shall obtain the general form of continuous linear functionals on some of the normed spaces we have encountered, as for Hilbert spaces in the “Little” Riesz representation theorem in Chapter 1. We prove two major representation theorems. The first describes the continuous linear functionals on the Lp-spaces, and the second, called the Riesz representation theorem, describes the continuous linear functionals on C0(X), the space of continuous complex-valued functions on a locally compact Hausdorff space X that vanish at infinity. We also construct the Haar measure on a locally compact topological group G, which is the (essentially unique) left translation invariant positive measure on G. Finally, the chapter offer exercises to challenge the reader.
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