Duality and distance formulas in spaces defined by means of oscillation (Q363501)

In this interesting article, the author develops a pattern to prove that a Banach space defined by some big-\(O\) condition is actually the bidual of the subspace given by the corresponding little-\(o\) condition: Let \(X,Y\) be Banach spaces, \(X\) separable and reflexive. Let \(\mathcal{L}\) be a subset of \(L(X,Y)\) endowed with a \(\sigma\)-compact locally compact topology such that, for every \(x\in X\), the evaluation mapping \(T_x: \mathcal{L}\to Y,\; T_x(L)=Lx,\) is continuous, and let \(\mathcal{L}\cup\{\infty\}\) be its one-point compactification. Consider the space \(M(X,\mathcal{L}):=\{x\in X: \sup_{L\in \mathcal{L}}\|Lx\|\text{ is finite}\}\) and its subspace \(M_0(X,\mathcal{L}):=\{x\in X: \limsup_{ \mathcal{L}\ni L\to \infty}\|Lx\|=0\}.\) Assume further that \(\|x\|_M:=\sup_{L\in \mathcal{L}}\|Lx\|\) turns \(M(X,\mathcal{L})\) into a Banach space that embeds continuously and densely in \(X.\)NEWLINENEWLINEThe main result is as follows: If, for every \(x\in M(X,\mathcal{L})\), there is a bounded sequence in \(M_0(X,\mathcal{L})\) that converges to \(x\) in \(X,\) then the inclusion map \(\iota:X^*\to M_0(X,\mathcal{L})^*\) is continuous, has dense range and its transpose map \(\iota^*\) is an isomorphism of \( M_0(X,\mathcal{L})^{**}\) onto \( M(X,\mathcal{L})\) whose restriction to \( M_0(X,\mathcal{L})\) is the identity. Moreover, if \(x\in M(X,\mathcal{L}),\) its distance to \(M_0(X,\mathcal{L})\) coincides with \(\limsup_{ \mathcal{L}\ni L\to \infty}\|Lx\|\). A condition for \(\iota^*\) to be an isometry is also given.NEWLINENEWLINEThe proof relies mainly on vector-valued integration and a representation theorem for the dual space of \(C_0(\mathcal{L},Y)\) due to \textit{I. Dobrakov} [Czech. Math. J. 21(96), 13--30 (1971; Zbl 0225.47018)].NEWLINENEWLINEIn the examples section, it is shown that a number of well-known spaces fall into the above scheme. Among them are the Bloch and little Bloch spaces, BMO and VMO spaces, BMOA and VMOA spaces, more general Möbius invariant spaces, weighted spaces \(H_{v}(\Omega)\) and \(H_{v_0}(\Omega),\) and others.