# [time 130] Re: [time 129] Re: [time 128] On Pratt's Duality

Sun, 4 Apr 1999 15:37:39 +0900

Dear Stephen,

The parts you quoted from Prugovecki are all well-known in math, and do not
reflect the special features of the dualism of Chu spaces.

Best,
Hitoshi

----- Original Message -----
From: Stephen P. King <stephenk1@home.com>
Pratt <pratt@CS.Stanford.EDU>
Sent: Sunday, April 04, 1999 1:58 PM
Subject: [time 129] Re: [time 128] On Pratt's Duality

> Dear Hitoshi,
>
> I don't think that he knows Pratt's work, but...
>
> page 33.
>
> "An M-coordinate-independent definition of 'covariant vectors'... can be
> obtained by introducing the *cotangent space* T_x^*M above x as the
> algebraic dual of T_xM. -i.e.as consisting of real-valued linear
> functionals w over T_xM. An equivalent definition of cotangent space can
> also be given in terms of the family of all smooth real-valued functions
> defined on some neighborhood N_x of x, which forms the basis of the
> definition (1.5), by introducing for each element f in that family the
> following linear maps:
>
> df: X |-> Xf \element R^1, X \element T_xM. (1.8)"
>
> ...
>
> Page 63. Note 11
> "Note that, in the case that g is a matrix that acts by matrix
> multiplication on the elements of R^n, for its action from the
> rightthose elements have to be viewed as one-row matrices, whereas for
> its action on the left they have to be viewed as one-column matrices, so
> that one mode of such action can be related to the other by taking the
> transposes of the matrices in question."
>
> These properties are consistent with a Chu space. I may have gotten a
> bit exited and missed something... :) There is much to cover and I am a
> bit tired. :)
>
> Later,
>
> Stephen
>
> >
> > Dear Stephen,
> >
> > ----- Original Message -----
> > From: Stephen P. King <stephenk1@home.com>
> > Cc: Time List <time@kitada.com>
> > Sent: Sunday, April 04, 1999 11:19 AM
> > Subject: [time 127] Re: [time 121] RE: [time 115] On Pratt's Duality
> >
> > > Dear Hitoshi,
> > >
> > > I apologize for the length of this... :) BTW, I think that Prugovecki's
> > > formalism already has Chu_2 spaces built in, he just does not understand
> > > the implications! More on this later... ;)
> >
> > At which points or where in the book does Prugovecki include Chu spaces?
> >
> > Best,
> > Hitoshi
>

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