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Osborne - Locally Convex Spaces

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Osborne Locally Convex Spaces
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For most practicing analysts who use functional analysis, the restriction to Banach spaces seen in most real analysis graduate texts is not enough for their research. This graduate text, while focusing on locally convex topological vector spaces, is intended to cover most of the general theory needed for application to other areas of analysis. Normed vector spaces, Banach spaces, and Hilbert spaces are all examples of classes of locally convex spaces, which is why this is an important topic in functional analysis. While this graduate text focuses on what is needed for applications, it also shows the beauty of the subject and motivates the reader with exercises of varying difficulty. Key topics covered include point set topology, topological vector spaces, the Hahn-Banach theorem, seminorms and Frchet spaces, uniform boundedness, and dual spaces. The prerequisite for this text is the Banach space theory typically taught in a beginning graduate real analysis course.

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M. Scott Osborne Graduate Texts in Mathematics Locally Convex Spaces 2014 10.1007/978-3-319-02045-7_1
Springer International Publishing Switzerland 2014
1. Topological Groups
M. Scott Osborne 1
(1)
Department of Mathematics, University of Washington, Seattle, WA, USA
Abstract
Every locally convex space is a topological group, that is, a group that is also a topological space in which the group operations (multiplication and inversion) are continuous. A large number of the most basic results about locally convex spaces are actually valid for any topological group and can be established in that context with only a little additional effort. Since topological groups are important in their own right, it seems worthwhile to establish these basic results in the context of topological groups.
1.1 Point Set Topology
Every locally convex space is a topological group, that is, a group that is also a topological space in which the group operations (multiplication and inversion) are continuous. A large number of the most basic results about locally convex spaces are actually valid for any topological group and can be established in that context with only a little additional effort. Since topological groups are important in their own right, it seems worthwhile to establish these basic results in the context of topological groups.
While the reader is assumed to be familiar with basic point set topology, there are some twists that may or may not be familiar. These are not so important for topological groups (though they are handy), but they are crucial for dealing with locally convex spaces.
A notational point should be made before proceeding. If A is a subset of a topological space, then its closure will be denoted by A . This is because we will need complex numbers, and Picture 1 will denote complex conjugation. Similarly, the interior of A will be denoted by int( A ). This is because A has traditionally been assigned a special meaning in the context of locally convex spaces (it is called the polar of A ).
There are basically three subjects to be discussed. The one most likely to already be familiar is the notion of a net. While locally convex spaces can be studied without this concept, some substitute (e.g., filters) would be necessary without them.
A net is basically a generalized sequence in which the natural numbers are replaced by a directed set.
Definition 1.1.
A directed set is a pair ( D , ), where D is a nonempty set and is a binary relation on D subject to the following conditions:
(i)
For all D , .
(ii)
For all , , D , and implies .
(iii)
For all , D , there exists D such that and .
Note that (i) and (ii) make D look like a partially ordered set; conspicuous by its absence is the antisymmetry condition. The lack of antisymmetry is important for a number of applications (see below) and does not affect things much. The crucial addition is condition (iii), which is what the word directed usually signifies.
For some reason, it has become traditional to denote the elements of a directed set with lowercase Greek letters.
Definition 1.2.
A net in a topological space X is a function from D to X , where ( D , ) is a directed set. This function is usually denoted by x (or y , or something similar). This net x converges to x if the following happens: Whenever U is an open subset of X , with x U , then there exists D such that
Note As usual we sometimes refer to the directed set as D rather than the - photo 2
Note: As usual, we sometimes refer to the directed set as D , rather than the more proper ( D , ). Similarly, means . Also, as above, the net is usually denoted by Locally Convex Spaces - image 3 or Locally Convex Spaces - image 4 , and convergence to x is denoted by x x , Locally Convex Spaces - image 5 , or Locally Convex Spaces - image 6 . Since the notion of a net is a generalization of the notion of a sequence (with Locally Convex Spaces - image 7 being replaced by D ), this is consistent with standard terminology for sequences.
Example 1 (cf. Bear []).
Given a bounded function Locally Convex Spaces - image 8 , the RiemannDarboux integral can be defined as a net limit as follows. A partition P of [ a , b ] is a finite sequence Locally Convex Spaces - image 9 , with Locally Convex Spaces - image 10 . That is, a partition is a finite set P , with Locally Convex Spaces - image 11 . A tagging T of the partition P above is a selection of points Locally Convex Spaces - image 12 for which Locally Convex Spaces - image 13 , and a tagged partition is an ordered pair ( P , T ) for which P is a partition and T is a tagging of P . The Riemann Sum S ( P , T , f ) for this tagged partition is the sum:
The directed set D is the set of all tagged partitions of a b with - photo 14
The directed set ( D , ) is the set of all tagged partitions of [ a , b ], with ( P , T ) ( P , T ) when P P . Note: The ordering ignores the tagging and so is not antisymmetric. Darbouxs version of the Riemann integral is defined as
See Bear for more details including how to produce Lebesgue integrals as a - photo 15
See Bear [] for more details, including how to produce Lebesgue integrals as a net limit.
The following three facts are elementary and provide typical examples of how the flexibility in choosing D can be exploited.
Proposition 1.3.
Suppose X is a topological space. Then:
(a)
If A X, then A is the set of limits of nets from A.
(b)
X is Hausdorff if, and only if, convergent nets have unique limits.
(c)
If Y is a topological space, and f: X Y is a function, then f is continuous if, and only if, for any net in X Proof a1 Suppose is a net in A and - photo 16 in X: Proof a1 Suppose is a net in A and If U is any open - photo 17 .
Proof.
(a1)
Suppose Locally Convex Spaces - image 18 is a net in A , and Locally Convex Spaces - image 19
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