# Topological Vector Spaces

@inproceedings{Bourbaki1987TopologicalVS, title={Topological Vector Spaces}, author={Nicolas Bourbaki}, year={1987} }

The general theory of topological vector spaces was founded during the period which goes from 1920 to 1930 approximately. But it had been prepared for a long time before by the study of numerous problems of Functional Analysis; and its history cannot be retraced without indicating, at least summarily, how the study of these problems gradually brought mathematicians (especially from the beginning of the XXth century) to become aware of the common ancestry of the questions under consideration… Expand

#### 220 Citations

Locally Convex Spaces

- Mathematics
- 2013

Once again, a topological vector space will be called a locally convex space if it is locally convex, that is, if each point has a neighborhood base consisting of convex sets. The Hausdorff condition… Expand

Pontryagin Duality in the Theory of Topological Vector Spaces and in Topological Algebra

- Mathematics
- 2003

The theory of topological vector spaces (TVS), being a foundation of modern functional analysis, is now considered as a completely mature, or, to be more specific, dead mathematical discipline. This… Expand

Subspaces with Equal Closure

- Mathematics
- 2001

Abstract
We take a new and unifying approach toward polynomial and trigonometric
approximation in
topological vector spaces used in analysis on Rn. The idea is to show in
considerable
generality… Expand

Differentiable mappings between spaces of sections

- Mathematics
- 2013

In this work, various versions of the so-called Omega-Lemma are provided, which ensure differentiability properties of pushforwrds between spaces of C^r-sections (or compactly supported C^r-sections)… Expand

Differential Calculus over General Base Fields and Rings

- Mathematics
- 2004

Abstract We present an axiomatic approach to finite- and infinite-dimensional differential calculus over arbitrary infinite fields (and, more generally, suitable rings). The corresponding basic… Expand

A characterization of Pontryagin–van Kampen duality for locally convex spaces☆

- Mathematics
- 2002

Abstract Topological vector spaces (TVSs) are topological Abelian groups when considered under the operation of addition. It is therefore natural to ask when they satisfy Pontryagin–van Kampen (P–vK)… Expand

Continuity of bilinear maps on direct sums of topological vector spaces

- Mathematics
- 2011

We prove a criterion for continuity of bilinear maps on countable direct sums of topological vector spaces. As a first application, we get a new proof for the fact (due to Hirai et al. 2001) that the… Expand

New Approach to Arakelov Geometry

- Mathematics
- 2007

This work is dedicated to a new completely algebraic approach to Arakelov geometry, which doesn't require the variety under consideration to be generically smooth or projective. In order to construct… Expand

The Space of Coefficients in a Linear Topological Space

- Mathematics
- 2012

It is proved that the arbitrary nondegenerate system in a linear complete topological space has a correspondence complete topological space of coefficients with canonical basis. Basicity criterion… Expand

On the PTAK Homomorphism Theorem

- Mathematics
- 1989

In this note, a brief and accessible proof is given of an extension of the Ptak homomorphism theorem to a larger class of spaces—spaces that are not necessarily assumed to be locally convex. This is… Expand