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Publisher Summary This chapter discusses normed spaces. The theory of normed spaces and its numerous applications and branches form a very extensive division of functional analysis. A normed ideal space (NIS) is an ideal space equipped with a monotone norm. An NIS which is an foundation space is called a normed foundation space (NFS). An NIS which is complete in its norm is called a Banach ideal space, while an NFS which is complete in its norm is called a Banach foundation space. The chapter also discusses Hilbert space. Hilbert space is an immediate generalization of Euclidean space, so its geometry approaches Euclidean geometry more closely than is the case for any other B-space. Hilbert space possesses many properties of Euclidean space that B-spaces in general do not possess. This situation has made it possible to develop functional analysis based on Hilbert space much more extensively and completely than functional analysis based on general normed spaces and, as a result, Hilbert space theory has separated off into a large independent branch of functional analysis with its own results and methods, not confined within the same limits as the general functional analysis. |
