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Real Analysis is the third volume in the Princeton Lectures in Analysis, a series of four textbooks that aim to present, in an integrated manner, the core areas of analysis. Here the focus is on the development of measure and integration theory, differentiation and integration, Hilbert spaces, and Hausdorff measure and fractals. The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean extends the methods of vector algebra and calculus from the two- dimensional Euclidean plane and three- dimensional space to spaces with any finite or infinite number of dimensions.
A Hilbert space is an abstract vector space possessing the structure of an inner product rmed vector spaces. The basic and historically first class of spaces studied in functional analysis are complete normed vector spaces over the real or complex numbers.
Such spaces are called Banach spaces. An important example is a Hilbert space, where the norm arises from an inner product.