Analysis Now

General or set-theoretical topology is the theory of continuity and convergence in analysis. Although the theory draws its notions and fundamental examples from geometry (so that the reader is advised always to think of a topological space as something resembling the euclidean plane), it applies most often to infinite-dimensional spaces of functions, for which geometrical intuition is very hard to obtain. Topology allows us to reason in these situations as if the spaces were the familier two- and three-dimensional objects, but the process takes a little time to get used to.

Assuming a basic knowledge of linear algebra we now infuse extra topological concepts and arrive at the theory of topological vector spaces. Keeping to the essentials we only develop the theory of locally convex spaces, and we make these appear as seminormed spaces (with initial topology).

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.

A function calculus for a Banach algebra U is a collection of algebra isomorphisms of the form Ф: b → U , where b is an algebra of continuous functions on some compact Hausdorff space X. Loosely speaking, a function calculus is deemed the better, the larger the function algebra b is inside C(X).

Many problems in analysis lead irrevocably to unbounded operators. It suffices to mention the differential process, for early encounters, and, as a branch of functional analysis, the theory of partial differential equations (the final showdown). This chapter does not, by a long shot, cover the theory of unbounded operators (and a good excuse would be that there is no theory, only myriads of examples). A small area of this vast territory—dealing with a single unbounded, self-adjoint (or, maybe, normal) operator in a Hilbert space—can, however, be cultivated by the spectral theory of bounded operators; and this we propose to do in some detail.

This chapter has two functions: Throughout the book it has served as an Appendix, to which the reader was referred for definitions, arguments, and results about measures and integrals. It will now serve as a functional analyst’s dream of the ideal short course in measure theory. Thus, we shall develop the theory of Radon integrals ( = Radon measures, cf. 6.3.4) on a locally compact Hausdorff space, assuming full knowledge of topology and topological vector spaces. This theory takes as point of departure an integral (a positive linear functional) on the minimal class of topologically relevant functions on X, namely, the class C _c (X) of continuous functions with compact supports. The integral is extended by monotonicity to a larger class of (integrable) functions and the measure appears, post festum, as the value of the integral on characteristic functions.