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

Stephen P. King (stephenk1@home.com)
Sat, 03 Apr 1999 23:58:36 -0500

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. :)



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
> > 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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