By using Hahn-Banach theorem, a characterization of random approximations is obtained.

was established by Hahn [1; p. 217] in 1927, and independently by Banach [2; p. 212] in 1929, who also generalized Theorem 0 for real spaces, to the situation in which the functional q :E^>R is an arbitrary subadditive, positive homogeneous functional [2; p. 226]. Theorem 0 was not established for complex spaces until 1938, when it was deduced from the real theorem by Bohnenblust and Sobczyk [3...

In this paper we discuss some fundamental results in real and functional analysis including the Riesz representation theorem, the Hahn-Banach theorem, and the Baire category theorem. We also discuss applications of these theorems to other topics in analysis.

in this paper, we first introduce the notion of $c$-affine functions for $c> 0$.then we deal with some properties of strongly convex functions in real inner product spaces by using a quadratic support function at each point which is $c$-affine. moreover, a hyers–-ulam stability result for strongly convex functions is shown.

We determine the computational complexity of the Hahn-Banach Extension Theorem. To do so, we investigate some basic connections between reverse mathematics and computable analysis. In particular, we use Weak König’s Lemma within the framework of computable analysis to classify incomputable functions of low complexity. By defining the multi-valued function Sep and a natural notion of reducibilit...

We determine the computational complexity of the Hahn-Banach Extension Theorem. To do so, we investigate some basic connections between reverse mathematics and computable analysis. In particular, we use Weak König’s Lemma within the framework of computable analysis to classify incomputable functions of low complexity. By defining the multi-valued function Sep and a natural notion of reducibilit...