In a first course in functional analysis, a great deal of time is spent with Banach spaces, especially the interaction between such spaces and their dual spaces. Banach spaces are a special type of topological vector space, and there are important topological vector spaces which do not lie in the Banach category, such as the Schwartz spaces. The most fundamental theorem about Banach spaces is t...

In the title salt, C6H10N3O2 (+)·C7H7O3S(-), the imidazole ring makes a dihedral angle of 70.93 (12)° with the plane of the toluene ring. In the crystal, the ions are linked via N-H⋯O and weak C-H⋯O hydrogen bonds forming two-dimensional networks lying parallel to (001). These networks are linked via C-H⋯π inter-actions, forming a three-dimensional structure.

In the history of the Rice Dramatic Club, there have been occasions when it has fallen in the power of the or ganization to produce outstanding modern dramas by contemporary playwrights—plays rich in enter ta inment quality, subtle in humor ; plays which live in the minds of the a u dience long af ter the final curtain falls. Such plays as "The Roynl Family," "Hay Fever ," and "Hell Bent for He...

Let 1 < p ̸= 2 < ∞, ε > 0 and let T : lp(l2) into → Lp[0, 1] be an isomorphism Then there is a subspace Y ⊂ lp(l2), (1 + ε)-isomorphic to lp(l2), such that: T|Y is an (1+ ε)-isomorphism and T (Y ) is Kp-complemented in Lp [0, 1], with Kp depending only on p. Moreover, Kp ≤ (1 + ε)γp if p > 2 and Kp ≤ (1 + ε)γp/(p−1) if 1 < p < 2, where γr is the Lr norm of a standard Gaussian variable.

Using a recent result of Batson, Spielman and Srivastava, We obtain a tight estimate on the dimension of lp , p an even integer, needed to almost isometrically contain all k-dimensional subspaces of Lp. In a recent paper [BSS] Batson, Spielman and Srivastava introduced a new method for sparsification of graphs which already proved to have functional analytic applications. Here we bring one more...