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The '''Radon–Riesz property''' is a mathematical property for [[normed space]]s that helps ensure [[limit of a sequence|convergence]] in norm. Given two assumptions (essentially weak convergence and continuity of norm), we would like to ensure convergence in the [[norm topology]].
 
==Definition==
Suppose that (''X'', ||·||) is a normed space. We say that ''X'' has the ''Radon–Riesz property'' (or that ''X'' is a ''Radon–Riesz space'') if whenever <math>(x_{n})</math> is a sequence in the space and <math>x</math> is a member of ''X'' such that <math>(x_{n})</math> [[weak topology|converges weakly]] to <math>x</math> and <math>\lim_{n\to\infty} \Vert x_n \Vert = \Vert x\Vert </math>, then <math>(x_{n})</math> converges to <math>x</math> in norm; that is, <math>\lim_{n\to\infty} \Vert x_n - x\Vert = 0 </math>.
 
==Other names==
 
Although it would appear that [[Johann Radon]] was one of the first to make significant use of this property in 1913, [[Mikhail Kadets|M. I. Kadets]] and V. L. Klee also used versions of the Radon–Riesz property to make advancements in [[Banach space]] theory in the late 1920s. It is common for the Radon–Riesz property to also be referred to as the '''Kadets–Klee property''' or '''property (H)'''. According to Robert Megginson, the letter H does not stand for anything. It was simply referred to as property (H) in a list of properties for normed spaces that starts with (A) and ends with (H). This list was given by K. Fan and I. Glicksberg. The "Riesz" part of the name refers to [[Frigyes Riesz]]. He also made some use of this property in the 1920s.
 
==Example==
Every real Hilbert space is a Radon–Riesz space. Indeed, suppose that ''H'' is a real [[Hilbert space]] and that <math> (x_{n}) </math> is a sequence in ''H'' converging weakly to a member <math> x </math> of ''H''. Using the two assumptions on the sequence and the fact that
:<math>\langle x_{n} - x, x_{n} - x\rangle = \langle x_{n} , x_{n} \rangle - \langle x_{n} , x\rangle  - \langle  x, x_{n} \rangle + \langle  x,  x\rangle, </math>
and letting ''n'' tend to infinity, we see that  
:<math>\lim_{n\to\infty}{\langle x_{n} - x, x_{n} - x\rangle} = 0. </math>
Thus ''H'' is a Radon–Riesz space.
 
==See also==
* [[Johann Radon]]
* [[Frigyes Riesz]]
* [[Hilbert space]] or [[Banach space]] theory
* [[Weak topology]]
* [[Normed space]]
* [[Functional analysis]]
* [[Schur's property]]
 
==Reference(s)==
* {{Citation
  | first = Robert E. | last = Megginson 
  | title = An Introduction to Banach Space Theory
  | place = New York Berlin Heidelberg
  | publisher = Springer-Verlag
  | year = 1998
  | isbn = 0-387-98431-3 }}
 
{{DEFAULTSORT:Radon-Riesz property}}
[[Category:Functional analysis]]

Revision as of 08:44, 15 January 2014

The Radon–Riesz property is a mathematical property for normed spaces that helps ensure convergence in norm. Given two assumptions (essentially weak convergence and continuity of norm), we would like to ensure convergence in the norm topology.

Definition

Suppose that (X, ||·||) is a normed space. We say that X has the Radon–Riesz property (or that X is a Radon–Riesz space) if whenever (xn) is a sequence in the space and x is a member of X such that (xn) converges weakly to x and limnxn=x, then (xn) converges to x in norm; that is, limnxnx=0.

Other names

Although it would appear that Johann Radon was one of the first to make significant use of this property in 1913, M. I. Kadets and V. L. Klee also used versions of the Radon–Riesz property to make advancements in Banach space theory in the late 1920s. It is common for the Radon–Riesz property to also be referred to as the Kadets–Klee property or property (H). According to Robert Megginson, the letter H does not stand for anything. It was simply referred to as property (H) in a list of properties for normed spaces that starts with (A) and ends with (H). This list was given by K. Fan and I. Glicksberg. The "Riesz" part of the name refers to Frigyes Riesz. He also made some use of this property in the 1920s.

Example

Every real Hilbert space is a Radon–Riesz space. Indeed, suppose that H is a real Hilbert space and that (xn) is a sequence in H converging weakly to a member x of H. Using the two assumptions on the sequence and the fact that

xnx,xnx=xn,xnxn,xx,xn+x,x,

and letting n tend to infinity, we see that

limnxnx,xnx=0.

Thus H is a Radon–Riesz space.

See also

Reference(s)

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