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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 $$L_ E^{*\phi}(\mu)$$.

##### MSC:
 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