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The set of L^p-functions (where p>=1) generalizes L2-space. Instead of square integrable, the measurable function f must be p-integrable for f to be in L^p. On a measure ...
A measure space is a measurable space possessing a nonnegative measure. Examples of measure spaces include n-dimensional Euclidean space with Lebesgue measure and the unit ...
Let X be a normed space and X^(**)=(X^*)^* denote the second dual vector space of X. The canonical map x|->x^^ defined by x^^(f)=f(x),f in X^* gives an isometric linear ...
A topological space is compact if every open cover of X has a finite subcover. In other words, if X is the union of a family of open sets, there is a finite subfamily whose ...
A half-space is that portion of an n-dimensional space obtained by removing that part lying on one side of an (n-1)-dimensional hyperplane. For example, half a Euclidean ...
A basis for the real numbers R, considered as a vector space over the rationals Q, i.e., a set of real numbers {U_alpha} such that every real number beta has a unique ...
A totally disconnected space is a space in which all subsets with more than one element are disconnected. In particular, if it has more than one element, it is a disconnected ...
A metric space X is boundedly compact if all closed bounded subsets of X are compact. Every boundedly compact metric space is complete. (This is a generalization of the ...
A G-space provides local notions of harmonic, hyperharmonic, and superharmonic functions. When there exists a nonconstant superharmonic function greater than 0, it is a ...
A topological space having a countable dense subset. An example is the Euclidean space R^n with the Euclidean topology, since it has the rational lattice Q^n as a countable ...
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