**Hitoshi Kitada** (*hitoshi@kitada.com*)

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

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Matti Pitkanen: "[time 131] Reply to Kitada"**Previous message:**Stephen P. King: "[time 129] Re: [time 128] On Pratt's Duality"**Next in thread:**Stephen P. King: "[time 141] Re: [time 130] On Pratt's Duality"

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>

To: Hitoshi Kitada <hitoshi@kitada.com>; Time List <time@kitada.com>; Vaughan

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
*

*>
*

*> Hitoshi Kitada wrote:
*

*> >
*

*> > Dear Stephen,
*

*> >
*

*> > ----- Original Message -----
*

*> > From: Stephen P. King <stephenk1@home.com>
*

*> > To: Hitoshi Kitada <hitoshi@kitada.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
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*> > > 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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