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Inductive limit of a sequence of balanced topological spaces in Orlicz spaces \(L_ E^{*\phi}(\mu)\). (English) Zbl 0607.46021
Let \(\phi\) and \(\psi\) be \(\phi\)-functions or Orlicz functions. We shall say that \(\phi\) increases essentially more rapidly than \(\psi\) for all u (for large u) in symbols \(\psi \ll \phi(\psi\ll^{l}\phi)\) if for \(c>0\) \[ \lim_{u\to 0}\frac{\psi(cu)}{\phi(u)}=0\quad and\quad \lim_{u\to \infty}\frac{\psi (cu)}{\phi(u)}=0\quad (\lim_{u\to \infty}\frac{\psi(cu)} {\phi(u)}=0). \] Let \(\phi\) be a \(\phi\)-function and let \((E,\Sigma,\mu)\) be a sequence with a positive measure. In [ibid. 23, 71-84 (1981)] we have considered some linear topology on \(L_ E^{*\phi}(\mu)\), denoted by \({\mathcal T}^{\ll \phi}\), which has a base of neighbourhoods of 0 consisting of all sets of the form: \(K_{\psi}(r)\cap L_ E^{*\phi}(\mu)\), where \(r>0\) and \(\psi\) is such that \(\psi \ll \phi.\)
Let \({\mathcal T}_ I\) be the topology on \(L_ E^{*\phi}(\mu)\) of the strict inductive limit of the sequence of balanced topological spaces \(((\bar K_{\phi}(2^ n)\), \({\mathcal T}_ 0|_{\bar K_{\phi}(2^ n)}):\) \(n\geq 0)\), where \({\mathcal T}_ 0\) is the topology of convergence in measure, i.e. \({\mathcal T}_ I\) is the finest of all linear topologies \({\mathcal T}\) on \(L_ E^{*\phi}(\mu)\) which satisfy the condition: \({\mathcal T}|_{\bar K_{\phi}(2^ n)}={\mathcal T}_ 0|_{\bar K_{\phi}(2^ n)}\) for \(n\geq 0.\)
The main aim of present paper is to prove that the topology \({\mathcal T}^{\ll \phi}\) is identical with the topology \({\mathcal T}_ I\) on \(L_ E^{*\phi}(\mu)\).

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46M40 Inductive and projective limits in functional analysis