In a first course in functional analysis, a great deal of time is spent with Banach spaces, especially the interaction between such spaces and their dual spaces. Banach spaces are a special type of topological vector space, and there are important topological vector spaces which do not lie in the Banach category, such as the Schwartz spaces. The most fundamental theorem about Banach spaces is t...

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...

In this article, we deal with dual spaces and the Hahn-Banach Theorem. At the first, we defined dual spaces of real linear spaces and proved related basic properties. Next, we defined dual spaces of real normed spaces. We formed the definitions based on dual spaces of real linear spaces. In addition, we proved properties of the norm about elements of dual spaces. For the proof we referred to de...

Paul Garrett [email protected] http://www.math.umn.edu/ g̃arrett/ [This document is http://www.math.umn.edu/ ̃garrett/m/fun/notes 2016-17/02 banach.pdf] 1. Basic definitions 2. Riesz’ Lemma 3. Counter-examples for unique norm-minimizing element 4. Normed spaces of continuous linear maps 5. Dual spaces of normed spaces 6. Banach-Steinhaus/uniform-boundedness theorem 7. Open mapping theorem 8. C...

Let T̃ = ∑n i=1 ũi⊗ vi : V → V = [v1, ..., vn] ⊂ X, where ũi ∈ V ∗ and X is a Banach space. Let T = ∑n i=1 ui ⊗ vi : X → V be an extension of T̃ to all of X (i.e., ui ∈ X∗) such that T has minimal (operator) norm. In this paper we show in particular that, in the case n = 2 and the field is R, there exists a rank-n T̃ such that ‖T‖ = ‖T̃‖ for all X if and only if the unit ball of V is either not smo...