Hitoshi Kitada (firstname.lastname@example.org)
Sun, 4 Apr 1999 15:37:39 +0900
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.
----- Original Message -----
From: Stephen P. King <email@example.com>
To: Hitoshi Kitada <firstname.lastname@example.org>; Time List <email@example.com>; Vaughan
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. :)
> Hitoshi Kitada wrote:
> > Dear Stephen,
> > ----- Original Message -----
> > From: Stephen P. King <firstname.lastname@example.org>
> > To: Hitoshi Kitada <email@example.com>
> > Cc: Time List <firstname.lastname@example.org>
> > 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
This archive was generated by hypermail 2.0b3 on Sun Oct 17 1999 - 22:31:51 JST