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Teichmüller's theorem asserts the existence and uniqueness of the extremal quasiconformal map between two compact Riemann surfaces of the same genus modulo an equivalence ...

The series sum_(j=1)^(infty)f_j(z) is said to be uniformly Cauchy on compact sets if, for each compact K subset= U and each epsilon>0, there exists an N>0 such that for all ...

A group acts freely if there are no group fixed points. A point which is fixed by every group element would not be free to move.

A proper subgroup is a proper subset H of group elements of a group G that satisfies the four group requirements. "H is a proper subgroup of G" is written H subset G. The ...

If a map f:G->G^' from a group G to a group G^' satisfies f(ab)=f(b)f(a) for all a,b in G, then f is said to be an antihomomorphism. Moreover, if G and G^' are isomorphic, ...

Let X be a locally convex topological vector space and let K be a compact subset of X. In functional analysis, Milman's theorem is a result which says that if the closed ...

Let a group G have a group presentation G=<x_1,...,x_n|r_j(x_1,...,x_n),j in J> so that G=F/R, where F is the free group with basis {x_1,...,x_n} and R is the normal subgroup ...

Calabi-Yau spaces are important in string theory, where one model posits the geometry of the universe to consist of a ten-dimensional space of the form M×V, where M is a four ...

In functional analysis, the Banach-Alaoglu theorem (also sometimes called Alaoglu's theorem) is a result which states that the norm unit ball of the continuous dual X^* of a ...

A group G is said to be finitely generated if there exists a finite set of group generators for G.