A Concise Overview of Functional Spaces in Modern Theory and Advanced Reasoning

<p>As we begin our exploration of Hilbert space the reader is assumed to have some background in linear algebra and real analysis. Nonetheless for the sake of clarity we begin with a discussion of three notions that are fundamental to the field of functional analysis namely metric spaces normed linear spaces and inner product spaces. Few definitions are as fundamental to analysis as that of the metric space. In essence a metric space is simply a collection of objects (e.g. numbers matrices pineapple flavored Bon Bons covered with flax seeds) with an associated rule or function that determines distance between two objects in the space. Such a function is termed a metric. Perhaps the most intuitive example of a metric space is the real number line with the associated metric |x ? y| for x y ? R. In general though a metric need only satisfy four basic criteria. More formally: Deftnition (Metric Space). A metric space (X d) is a set X together with an assigned metric function d : X × X ? R that has the following properties: Positive: d(x y) ? 0 for all x y z ? X</p>