Two notions of tangent vector for a Fréchet manifold - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T11:22:51Z http://mathoverflow.net/feeds/question/43202 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/43202/two-notions-of-tangent-vector-for-a-frechet-manifold Two notions of tangent vector for a Fréchet manifold David Carchedi 2010-10-22T16:25:54Z 2010-10-23T07:41:19Z <p>Let $X$ be a Frechet or Banach manifold. We can define tangent vectors by equivalence classes of smooth curves. But, we could also define them as derivations of germs of smooth functions. Do these two notions agree in this infinite dimensional setting?</p> http://mathoverflow.net/questions/43202/two-notions-of-tangent-vector-for-a-frechet-manifold/43205#43205 Answer by AndrĂ© Henriques for Two notions of tangent vector for a Fréchet manifold AndrĂ© Henriques 2010-10-22T16:48:49Z 2010-10-22T16:48:49Z <p>Let $X$ be a Banach or Fr&eacute;chet vector space.<br> Then the following two notions already don't agree:<br> 1) linear map $\mathbb R\to X$<br> 2) linear functional on the space of linear functions.</p> http://mathoverflow.net/questions/43202/two-notions-of-tangent-vector-for-a-frechet-manifold/43208#43208 Answer by Gjergji Zaimi for Two notions of tangent vector for a Fréchet manifold Gjergji Zaimi 2010-10-22T16:58:50Z 2010-10-22T16:58:50Z <p>I believe the condition for these two definition to agree is that the Frechet manifold be nuclear. See section 28 (proposition 28.7) of "The convenient setting of global analysis" by A. Kriegl and P.W. Michor.</p> http://mathoverflow.net/questions/43202/two-notions-of-tangent-vector-for-a-frechet-manifold/43270#43270 Answer by Andrew Stacey for Two notions of tangent vector for a Fréchet manifold Andrew Stacey 2010-10-23T07:41:19Z 2010-10-23T07:41:19Z <p>To expand on Gjergji's answer a little:</p> <p>According to Kriegl and Michor, the key for the identification is two properties on the <strong>model</strong> space:</p> <ol> <li>It be reflexive</li> <li>It have the <em>bornological approximation property</em></li> </ol> <p><strong>Reflexivity</strong> is needed because, essentially, of what Andr&eacute; mentions: linear maps in to a space are not always the same as linear maps out of linear maps out of the space! The derivational definition of tangent vectors is as operators on cotangent vectors (even if dressed up a derivations on germs) so it's a "dual of a dual" and you need reflexivity to get back where you started.</p> <p><strong>Approximation</strong> properties (of which there are many) are a way of saying that linear maps from one space to another aren't too messy, by which we mean that they behave a little like matrices. Matrices themselves correspond to things in $E \otimes E'$, but as we're in infinite dimensions then that's only finite rank operators. Still, as we're Grown Ups, we can handle approximations and so it's enough for us that any linear transformation can be approximated by a matrix. With the strong (or operator) topology, this only gets us compact operators, but with a weak topology we have a chance of getting the whole thing. The different types of approximation property correspond to different topologies on linear maps.</p> <p>There are spaces (even Banach spaces) that don't have the requisite approximation property, but they're "not nice" spaces and also not ones that are usually encountered. All the "most common" spaces have the approximation property.</p> <p>In particular, nuclear Frechet spaces are reflexive and have the approximation property, so - as Gjerji says - that's sufficient. However, Hilbert spaces are also reflexive and have the approximation property.</p> <p>The last thing I have to say on this is that if your <em>real</em> question is "What's the right notion of tangent vector in infinite dimensions?" then you should look at the chart in Kriegl and Michor's book in section 33.21 (p351 in the current version online). From that chart, it's clear that the "right" definition is as equivalence classes of curves.</p>