A note on norm-one projections onto subspaces of finite codimension of \(l^{\infty}\)
Let X be a real linear normed space and let \(Y\subset X\) be a closed subspace. An operator P: \(X\to Y\) is a projection if P is linear, continuous and such that \(Py=y\) for any y in Y. A projection P is said to be: 1) strictly contractive if \(\| Px\| <\| x\|\) for any \(x\not\in Y,\) 2) bicontractive if \(\| P\| =\| I-P\| =1.\) In this paper we prove that in \(\ell^{\infty}\) no subspace of finite codimension is the range of a strictly contractive projection and moreover we characterize the hyperplanes which are the range of a bicontractive projection. Some references about projections are in \textit{M. Baronti} and \textit{P. L. Papini} [Ann. Mat. Pura Appl. 4, 53-61 (1988)], \textit{B. Beauzamy} [Bull. Sci. Math. 102, 43-47 (1978; Zbl 0376.46011)].
scientific article; zbMATH DE number 4117215