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Normal Structure, Fixed Point and Some Related Parameters in Banach Spaces

Who: Dr. Ji Gao, Community College of Phil.

When and Where: Thursday, November 20, in Barbelin 226 at 11:50.

Food? Yes, sandwiches, chips, and soda. Food will be available prior to the start of the colloquium in BL 226 beginning at 11:30. Anyone attending the talk is welcome.

Audience: Math faculty and upper-level students (analysis would be helpful).

Abstract:

Following a brief introduction to Banach Spaces, the following will be shown. Let $X$ be a Banach space, $X_{2} \subseteq X$ be a two dimensional subspace of $X$, and $S(X) = \{x \in X , \Vert x\Vert = 1\}$ be the unit sphere of $X$. Some Parameters, $K_{\xi,\eta}(X) = sup\{\Vert \xi x + \eta y \Vert+ \Vert \xi x - \eta y \Vert, x \in S(X), y \in S(X)\}$ where $\xi, \eta>0$, the modulus of $W^{*}$-convexity, $W^{*}(\epsilon)=inf\{< \frac{x-y}{2}, f_{x}>: x, y\in S(X), \Vert x-y\Vert\geq \epsilon, f_{x}\in \nabla_{x}\}$, where $0\leq \epsilon \leq 2$ and $\nabla _{x}\subseteq S(X^{*})$ be the set of norm 1 supporting functionals of $S(X)$ at $x$, and others are introduced and studied. Let $\rho_{X}(\epsilon):[0,+\infty ) \rightarrow [ 0, +\infty )$ be the modulus of smoothness of X. The main results are that a Banach space X with $K_{\xi,\eta}(X) < 2max\{\xi,\eta\} + min\{\xi,\eta\}$ has uniform normal structure; and a Banach space X with $\rho_{X}(\epsilon) < \frac{\epsilon}{2}$ for some $0 \leq \epsilon \leq 1$, or $\rho_{X}(\epsilon) < \epsilon - \frac{1}{2}$ for some $\epsilon \geq 1$ has uniform normal structure. The relationship between normal structure and the arc length in $X$ is studied. Let $R(X) = inf \{l(S(X_{2}))- r(X_{2}): X_{2} \subseteq X \}$, where $l(S(X_{2}))$ is the circumference of $S(X_{2})$ and $r(X_{2})= sup \{2(\vert\vert x + y\vert\vert + \vert\vert x - y\vert\vert): x, y \in S(X_{2})\}$ is the least upper bound of the perimeters of the inscribed parallelogram of $S(X_{2})$. Then $R(X) > 0$ implies $X$ has uniform normal structure.
Several papers by Dr. Gao are available MaryAnne's office in the Mail Cubbies.

The next colloquium is scheduled for January to be given by Saint Joseph's Visiting Faculty Member Greg Naber. He will be speaking in November on Topology, Geometry and Physics: The Witten Conjecture.

Presented by the SJU Math and Computer Science Department.

Sean Forman and Jonathan Hodgson, colloquium committee