the Weierstrass-Casorati theorem, Laurent series, and the classification of singularities as removable, poles and essential singularities) generalize to results on harmonic functions in any dimension. By considering which theorems of complex analysis are special cases of theorems of potential theory in any dimension, one can obtain a feel for exactly what is special about complex analysis in two dimensions and what is simply the two-dimensional instance of more general results. Local behavior An important topic in potential theory is the study of the local behavior of harmonic functions. Perhaps the most fundamental theorem about local behavior is the regularity theorem for Laplace's equation, which states that harmonic functions are analytic. There are results which describe the local structure of level sets of harmonic functions. There is Bôcher's theorem, which characterizes the behavior of isolated singularities of positive harmonic functions. As alluded to in the last section, one can classify the isolated singularities of harmonic functions as removable singularities, poles, and essential singularities. Inequalities A fruitful approach to the study of harmonic functions is the consideration of inequalities they satisfy. Perhaps the most basic such inequality, from which most other inequalities may be derived, is the maximum principle. Another important result is Liouville's theorem, which states the only bounded harmonic functions defined on the whole of Rn are, in fact, constant functions. In addition to these basic inequalities, one has Harnack's inequality, which states that positive harmonic functions on bounded domains are roughly constant. One important use of these inequalities is to prove convergence of families of harmonic functions or sub-harmonic functions, see Harnack's theorem. These convergence theorems are used to prove the existence of harmonic functions with particular properties. Spaces of harmonic functions Since the Laplace equation is linear, the set of harmonic functions defined on a given domain is, in fact, a vector space. By defining suitable norms and/or inner products, one can exhibit sets of harmonic functions which form Hilbert or Banach spaces. In this fashion, one obtains such spaces as the Hardy space, Bloch space, Bergman space and Sobolev space. See also Subharmonic function Kellogg's theorem References S. Axler, P. Bourdon, W. Ramey (2001). Harmonic Function Theory (2nd edition). Springer-Verlag. . O. D. Kellogg (1969). Foundations of Potential Theory. Dover Publications. . L. L. Helms (1975). Introduction to potential theory. R. E. Krieger . J. L. Doob. Classical Potential Theory and Its Probabilistic Counterpart, Springer-Verlag, Berlin Heidelberg