Smoothly embedded subspaces of a Banach space
Let \(Y\) be a closed subspace of a real Banach space \(X\). If every continuous linear functional \(g\in Y^*\) has a unique norm-preserving extension \(f\in X^*\), then \(Y\) is called Hahn-Banach smooth in \(X\) (or \(Y\) is said to have (Phelps's) property \(U\) in \(X\)). The authors say that \(Y\) smoothly embeds into \(X\) (or \(Y\) is a totally smooth subspace of \(X\)) if every closed subspace \(Z\) of \(Y\) is Hahn-Banach smooth in \(X\). By the Taylor-Foguel theorem, \(Y\) smoothly embeds into \(X\) if and only if \(Y\) is Hahn-Banach smooth in \(X\) and the dual space \(Y^*\) is strictly convex. The authors prove that \(Y\) smoothly embeds into \(X\) if and only if \(Y\) has the following half-space property in \(X\): For every sequence \(B_n=B(y_n, r_n)\), \(n=1,2,\dots\), of open balls in \(X\) with centers \(y_1,y_2,\dots\) in \(Y\) and of radii \(r_n\to\infty\) such that \(B_1\subset B_2\subset\dots\), the union \(\bigcup^\infty_{n=1}B_n\) is the whole space \(X\) or an open half-space. The special case of this result when \(Y=X\) is known as Vlasov's theorem.
scientific article; zbMATH DE number 5885187