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 Eurasian Math. J., 2012, Volume 3, Number 4, Pages 44–52 (Mi emj104)

Orthogonality and smooth points in $C(K)$ and $C_b(\Omega)$

D. J. Kečkić

Abstract: For the usual norm on spaces $C(K)$ and $C_b(\Omega)$ of all continuous functions on a compact Hausdorff space $K$ (all bounded continuous functions on a locally compact Hausdorff space $\Omega$), the following equalities are proved:
$$\lim_{t\to0+}\frac{\|f+tg\|_{C(K)}-\|f\|_{C(K)}}t=\max_{x\inż\mid |f(z)|=\|f\|\}}\operatorname{Re}(e^{-i\arg f(x)}g(x))$$
and
$$\lim_{t\to0+}\frac{\|f+tg\|_{C_b(\Omega)}-\|f\|_{C_b(\Omega)}}t=\inf_{\delta>0}\sup_{x\inż\mid |f(z)|\ge\|f\|-\delta\}}\operatorname{Re}(e^{-i\arg f(x)}g(x)).$$
These equalities are used to characterize the orthogonality in the sense of James (Birkhoff) in spaces $C(K)$ and $C_b(\Omega)$ as well as to give necessary and sufficient conditions for a point on the unit sphere to be a smooth point.

Keywords and phrases: orthogonality in the sense of James, Gateaux derivative, smooth points.

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Bibliographic databases:
MSC: 46G05, 46E15, 49J50
Language:

Citation: D. J. Kečkić, “Orthogonality and smooth points in $C(K)$ and $C_b(\Omega)$”, Eurasian Math. J., 3:4 (2012), 44–52

Citation in format AMSBIB
\Bibitem{Kec12}
\by D.~J.~Ke{\v{c}}ki{\'c}
\paper Orthogonality and smooth points in $C(K)$ and $C_b(\Omega)$
\jour Eurasian Math. J.
\yr 2012
\vol 3
\issue 4
\pages 44--52
\mathnet{http://mi.mathnet.ru/emj104}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3040686}
\zmath{https://zbmath.org/?q=an:1281.46015}