 CARMA SEMINAR
 Speaker: Dr Riki Brown, University College London and University of Newcastle Upon Tyne
 Title: Metric Projections in Spaces of Continuous Functions
 Location: Room V129, Mathematics Building (Callaghan Campus) The University of Newcastle
 Time and Date: 3:00 pm, Wed, 15^{th} Jun 2011
 Abstract:
Let $T$ be a topological space (a compact subspace of ${\mathbb R^m}$, say) and let $C(T)$ be the space of real continuous functions on $T$, equipped with the uniform norm: $f = \text{max}_{t\in T}f(t)$ for all $f \in C(T)$. Let $G$ be a finitedimensional linear subspace of $C(T)$. If $f \in C(T)$ then
$$d(f,G) = \text{inf}\{f−g : g \in G\}$$
is the distance of $f$ from $G$, and
$$P_G(f) = \{g \in G : f−g = d(f,G)\}$$
is the set of best approximations to $f$ from $G$. Then
$$P_G : C(T) \rightarrow P(G)$$
is the setvalued metric projection of $C(T)$ onto $G$. In the 1850s P. L. Chebyshev considered $T = [a, b]$ and $G$ the space of polynomials of degree $\leq n − 1$. Our concern is with possible properties of $P_G$. The historical development, beginning with Chebyshev, Haar (1918) and Mairhuber (1956), and the present state of knowledge will be outlined. New results will demonstrate that the story is still incomplete.
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