LOCAL COMPLETENESS, LOWER SEMI CONTINUOUS FROM ABOVE FUNCTIONS AND EKELAND'S PRINCIPLE

Computability on Continuou, Lower Semi-continuous and Upper Semi-continuous Real Functions

In this paper we investigate continuous and upper and lower semi-continuous real functions in the framework of TTE, Type-2 Theory of EEectivity. First some basic facts about TTE are summarized. For each of the function spaces, we introduce several natural representations based on diierent intiuitive concepts of \eeectivity" and prove their equivalence. Computability of several operations on the...

Convex extensions and envelopes of lower semi-continuous functions

We define a convex extension of a lower-semicontinuous function to be a convex function that is identical to the given function over a pre-specified subset of its domain. Convex extensions are not necessarily constructible or unique. We identify conditions under which a convex extension can be constructed. When multiple convex extensions exist, we characterize the tightest convex extension in a...

On the Epi - Compactness Ofequi - Lower Semi - Continuous Functions

In this paper we show that equi-lsc. functions from a topological vector space X to the extended reals are epi-compact without assuming the local compactness or the second countablity of the underlying space X. We also show that weakly equi-lower semicontinuous functions from a Banach space X to the extended reals are Mosco-compact. Finally, we apply these results to prove the Mosoc-compactness...

Completeness Theorem for Continuous Functions and Product Class-topologies

We introduce an infinitary logic LA(On, Cn)n∈ω which is an extension of LA obtained by adding new quantifiers On and C, for every n ∈ ω. The corresponding models are topological class-spaces. An axiomatization is given and the completeness theorem is proved.

A Theorem on Semi-Continuous Functions