Polynomially reflexive space | Wikipedia audio article
Channel: wikipedia tts
Date: 2019-03-28

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Polynomially_reflexive_space


00:01:10 1 Relation to continuity of forms
00:02:06 2 Examples
00:02:57 3 Notes



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SUMMARY
=======
In mathematics, a polynomially reflexive space is a Banach space X, on which the space of all polynomials in each degree is a reflexive space.
Given a multilinear functional Mn of degree n (that is, Mn is n-linear), we can define a polynomial p as

  
    
      
        p
        (
        x
        )
        =
        
          M
          
            n
          
        
        (
        x
        ,
        …
        ,
        x
        )
      
    
    {\displaystyle p(x)=M_{n}(x,\dots ,x)}
  (that is, applying Mn on the diagonal) or any finite sum of these. If only n-linear functionals are in the sum, the polynomial is said to be n-homogeneous.
We define the space Pn as consisting of all n-homogeneous polynomials.
The P1 is identical to the dual space, and is thus reflexive for all reflexive X. This implies that reflexivity is a prerequisite for polynomial reflexivity.