TY - GEN

T1 - MR2543732 (2010g:46038) Colao, Vittorio; Trombetta, Alessandro; Trombetta, Giulio Hausdorff norms of retractions in Banach spaces of continuous functions. Taiwanese J. Math. 13 (2009), no. 4, 1139–1158. (Reviewer: Diana Caponetti)

AU - Caponetti, Diana

PY - 2009

Y1 - 2009

N2 - A retraction $R$ from the closed unit ball of aBanach space $X$ onto its boundary is called $k$-ball contractiveif there is $k \ge 0$ such that $ \gamma_X(RA) \le k \gamma_X(A)$ for each subset $ A$ of the closed unit ball, where $\gamma_X$ denote the Hausdorff (ball) measure of noncompactness. In the paper under review the authors consider the problem of evaluating the Wo\'{s}ko constant, which is the infimum of all numbers $k$'s for which there is a$k$-ball contractive retraction from the closed unit ball onto the sphere, in Banach spaces of real continuous functions defined on domains which are not necessarily bounded or finite dimensional.The paper extends some previous results valid in spaces ofcontinuous functions to a more general setting. The authorsconsider the space $\mathcal{B}\mathcal{C}_{B(E)}(K)$ of all realbounded functions which are continuous on $K$ and uniformlycontinuous on the closed unit ball $B(E)$, being $E$ a normedspace and $K$ a set in $E$ containing $B(E)$. They also considerthe space ${\mathcal C}(P)$ of all real continuous functionsdefined on the Hilbert cube $P= \{ x=(x_n) \in l_2 : |x_n| \le\frac1{n} \ \ (n=1,2, ...) \}$. They prove that in both thespaces $\mathcal{B}\mathcal{C}_{B(E)}(K)$ and ${\mathcal C}(P)$the Wo\'{s}ko constant assumes the smallest possible value $1$,they also give precise estimates of the lower Hausdorff norms andthe Hausdorff norms of the retractions they construct.

AB - A retraction $R$ from the closed unit ball of aBanach space $X$ onto its boundary is called $k$-ball contractiveif there is $k \ge 0$ such that $ \gamma_X(RA) \le k \gamma_X(A)$ for each subset $ A$ of the closed unit ball, where $\gamma_X$ denote the Hausdorff (ball) measure of noncompactness. In the paper under review the authors consider the problem of evaluating the Wo\'{s}ko constant, which is the infimum of all numbers $k$'s for which there is a$k$-ball contractive retraction from the closed unit ball onto the sphere, in Banach spaces of real continuous functions defined on domains which are not necessarily bounded or finite dimensional.The paper extends some previous results valid in spaces ofcontinuous functions to a more general setting. The authorsconsider the space $\mathcal{B}\mathcal{C}_{B(E)}(K)$ of all realbounded functions which are continuous on $K$ and uniformlycontinuous on the closed unit ball $B(E)$, being $E$ a normedspace and $K$ a set in $E$ containing $B(E)$. They also considerthe space ${\mathcal C}(P)$ of all real continuous functionsdefined on the Hilbert cube $P= \{ x=(x_n) \in l_2 : |x_n| \le\frac1{n} \ \ (n=1,2, ...) \}$. They prove that in both thespaces $\mathcal{B}\mathcal{C}_{B(E)}(K)$ and ${\mathcal C}(P)$the Wo\'{s}ko constant assumes the smallest possible value $1$,they also give precise estimates of the lower Hausdorff norms andthe Hausdorff norms of the retractions they construct.

KW - Measure of noncompactness

KW - Measure of noncompactness

UR - http://hdl.handle.net/10447/55815

M3 - Other contribution

ER -