Hilbert Spaces

Abstract

The geometry of infinite-dimensional Banach spaces offers quite a few surprises from the viewpoint of finite-dimensional euclidean spaces. Thus, the unit ball may have corners, and closed convex sets may fail to have elements of minimal norm. Even more alienating, there may be no notion of perpendicular vectors and no good notion of a basis. By contrast, the Hilbert spaces are perfect generalizations of euclidean spaces, to the point of being almost trivial as geometrical objects. The deeper theory (and the fruitful applications) is, however, concerned with the operators on Hilbert space. Accordingly, we devote only a single section to Hilbert spaces as such, centered around the notions of sesquilinear forms, orthogonality, and self-duality. We then develop the elementary theory of bounded linear operator on a Hilbert space ℌ, i.e. we initiate the study of the Banach *-algebra B(ℌ)—to be continued in later chapters.