A Banach space is polynomially Schur if sequential convergence against analytic polynomials implies norm convergence. Carne, Cole and Gamelin show that a space has this property and the Dunford-Pettis property if and only if it is Schur. Herein is defined a reasonable generalization of the Dunford-Pettis property using polynomials of a fixed homogeneity. It is shown, for example, that a Banach space will has the P N Dunford-Pettis property if and only if every weakly compact N −homogeneous polynomial (in the sense of Ryan) on the space is completely continuous. A certain geometric condition, involving estimates on spreading models and implied by nontrivial type, is shown to be sufficient to imply that a space is polynomially Schur.
Abstract. We consider the space of all Lipschitz functions on a metric space with bounded Lipschitz norm, and give an intrinsic characterization of the extreme points of the unit ball. We briefly discuss some examples of extreme Lipschitz functions, and apply the result to show that if the norm of a Banach space is Gateaux differentiable then extreme functions on any one-dimensional subspace may be canonically extended to extreme functions on the whole space.
The paper presents the guiding ideas behind our culturally responsive approach to teacher professional development and an overview of how those tenets inform, tacitly and directly, our efforts to realize the promise of the National Council of Teachers of Mathematics' five Process Standards. A review of the primary obstacles teachers face in implementing these standards in their own teaching and learning is followed by a description of the design elements in a university-based professional development program. Our goal is to provide an example of the foundations upon which an evolving approach to culturally responsive professional development planning has grown. We discuss research on what constitutes effective teacher professional development while noting the paucity of programs that embrace recognized needs. We do not give a prescription for effective teacher development. Instead, we speak as teacher-educators about the necessary philosophical and self-evaluative underpinnings to effective professional development and our approach to creating an environment where it is safe to leave the isolation of forced autonomy and it is valued to be reflective about community, mathematical activity, and intellectual engagement.
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