Hausdorff Norms of Retractions in Banach Spaces of Continuous Functions

Abstract

We construct retractions with positive lower Hausdorff norms and small Hausdorff norms in Banach spaces of real continuous functions which domains are not necessarily bounded or finite dimensional. Moreover, we give precise formulas for the lower Hausdorff norms and the Hausdorff norms of such maps.

Optimal retraction problem for proper $k$-ball-contractive mappings in $C^m [0,1]$

“…Concerning general results in the setting of Banach spaces, in [27] it was proved that W γ (X) ≤ 6 for any Banach space X, reaching the value 4 or 3 depending on the geometry of the space. Moreover it has been proved that W γ (X) = 1 in some spaces of continuous functions ( [7], [15]), in some classical Banach spaces of measurable functions ( [12]) and in Banach spaces whose norm is monotone with respect to some basis ( [3]). In [10] the problem of evaluating the Wośko constant has been considered in the setting of F -normed spaces.…”