**Stephen P. King** (*stephenk1@home.com*)

*Sat, 03 Apr 1999 23:58:36 -0500*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Hitoshi Kitada: "[time 130] Re: [time 129] Re: [time 128] On Pratt's Duality"**Previous message:**Hitoshi Kitada: "[time 128] Re: [time 127] Re: [time 121] RE: [time 115] On Pratt's Duality"**In reply to:**Hitoshi Kitada: "[time 121] RE: [time 115] On Pratt's Duality [time 81] Entropy, wholeness, dialogue, algebras"**Next in thread:**Hitoshi Kitada: "[time 130] Re: [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
*

**Next message:**Hitoshi Kitada: "[time 130] Re: [time 129] Re: [time 128] On Pratt's Duality"**Previous message:**Hitoshi Kitada: "[time 128] Re: [time 127] Re: [time 121] RE: [time 115] On Pratt's Duality"**In reply to:**Hitoshi Kitada: "[time 121] RE: [time 115] On Pratt's Duality [time 81] Entropy, wholeness, dialogue, algebras"**Next in thread:**Hitoshi Kitada: "[time 130] Re: [time 129] Re: [time 128] On Pratt's Duality"

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