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A "pointwise-bounded" family of continuous linear operators from a Banach space to a normed space is "uniformly bounded." Symbolically, if sup||T_i(x)|| is finite for each x ...

Several flavors of the open mapping theorem state: 1. A continuous surjective linear mapping between Banach spaces is an open map. 2. A nonconstant analytic function on a ...

A notation for large numbers defined by Steinhaus (1983, pp. 28-29). In this notation, denotes n^n, denotes "n in n triangles," and denotes "n in n squares." A modified ...

Szemerédi's theorem states that every sequence of integers that has positive upper Banach density contains arbitrarily long arithmetic progressions. A corollary states that, ...

If A is a unital Banach algebra where every nonzero element is invertible, then A is the algebra of complex numbers.

An "area" which can be defined for every set--even those without a true geometric area--which is rigid and finitely additive.

Let A be a closed convex subset of a Banach space and assume there exists a continuous map T sending A to a countably compact subset T(A) of A. Then T has fixed points.

A normed vector space X=(X,||·||_X) is said to be uniformly convex if for sequences {x_n}={x_n}_(n=1)^infty, {y_n}={y_n}_(n=1)^infty, the assumptions ||x_n||_X<=1, ...

A Tauberian theorem is a theorem that deduces the convergence of an series on the basis of the properties of the function it defines and any kind of auxiliary hypothesis ...

Any bounded planar region with positive area >A placed in any position of the unit square lattice can be translated so that the number of lattice points inside the region ...