Re: Partitions of unity proof roadblock

On Dec 1, 8:28 am, Jeff Rubin <JeffBRu...@xxxxxxxxx> wrote:
On Nov 30, 5:21 pm, smn <smnewber...@xxxxxxxxxxx> wrote:





On Nov 30, 4:27 pm, Jeff Rubin <JeffBRu...@xxxxxxxxx> wrote:

On Nov 30, 3:08 pm, smn <smnewber...@xxxxxxxxxxx> wrote:

On Nov 30, 10:59 am, Jeff Rubin <JeffBRu...@xxxxxxxxx> wrote:

I recently posted about the trouble I was having understanding
a proof in Lang's Fundamentals of Differential Geometry relating
to partitions of unity. So I tried turning to another text,
Manifolds, Tensor Analysis, and Applications by Abraham,
Marsden, and Ratiu. This text had a different approach but
my understanding ran into the same roadblock. I was wondering
if someone could help me out and explain.

The proof is of proposition 5.5.17 where M is a paracompact manifold
modeled on a Banach space E. It shows the equivalence of a number
of statements, the last two of which are:
(iv) every chart domain of M admits C^k partitions of unity
subordinate
to any open covering
(v) E admits C^k partitions of unity subordinate to any open covering
of E.

I had no problem with the implications (i)=>(ii)=>(iii)=>(iv). But
the proof
of (iv)=>(v) has got me stumped:

I wanted to describe a simplified version of the proof which pointed
out the problem I'm having, but I can't figure out how to do that
without sounding very confused. So please bear with me; here
is the full proof directly from the text: (I use /\ for intersection
and \/ for union)

"Consider now any open covering {U_\alpha}_{\alpha \in A} of E and let
(U, \phi) be an arbitrary chart of M. Refine first the covering of E
by taking
the intersections of all its elements with all translates of \phi(U).
Since E
is paracompact, refine again to a locally finite open covering {V_
\beta}.

When you do this you can refine by open balls whose closure B in E
(the closed balls )are still
contained in U .The function g of the partition of unity below that
worries you is 0 on E\B (open in E ) and C^k on U (open in E) and
hence C^k in E

Regards,smn

The inverse images by translations and \phi of these open sets are
subsets of U,
hence chart domains, and thus by (iv) they admit partitions of unity
subordinate
to any open covering, for example to {V_\beta /\ U_\alpha : \alpha \in
A};
call it {g_i^\beta}. Then g = \sum_{i,\beta} g_i^\beta is a C^k map
and the
double-indexed set of functions g_i^\beta / g forms a C^k partitions
of unity
of E."

Now I follow this down to and including that {g_i^\beta} is a
partition of unity
subordinate to the mentioned covering of V_\beta. Now these functions
g_i^\beta are necessarily C^k maps from V_\beta into R. They are not
(yet) maps from E into R. To get that you would have to extend them
by assigning the value 0 to all the points of E - V_\beta. You then
have
to show that the resulting extension is C^k. This is fine on points
in V_\beta
and on points outside of the closure of V_\beta, but gives me trouble
on
the boundary of V_\beta. There is no guarantee that the locally
finite covering
of V_\beta (that comes with the partition of unity {g_i^\beta}) is
locally finite
with respect to any points outside of V_\beta.

How do you fix this?

Thanks,

Jeff

Thanks for responding. I know what you are getting at, but the
details
still leave me puzzled. I am familiar with the result that in a
normal
space you can shrink a locally finite open covering of a set to one
whose
closures are contained in the corresponding open sets of the original
covering.
In the proof given in the cited book, it is not clear where doing this
shrinking helps. Once you have a manifold domain and a covering, you
get a partition of unity subordinate to that covering;
that is, a locally finite open refinement and the C^k functions which
sum to one and whose supports
lie in the locally finite open covering. If you try to shrink the
covering
that is part of the partition of unity, then what do you do to the
functions
to regain the support requirement? I'm still so confused.

Jeff- Hide quoted text -

- Show quoted text -

The point is to get the closures IN E (not just in U) to be in the
cordinate domain U .I think you can do that in the original
construction in U ,using balls as I said .No shrinking theorem used
later .Regards smn

Okay. I can see how to get the g_i^\beta to extend to
all of E and still be C^k. Thanks for that bit of the
puzzle. But I am immediately stymied by the next step,
which is to form the double sum over i and \beta of
g_i^\beta and to show that is C^k. For each point in E,
I need to find a neighborhood of the point which
intersects at most finitely many of the carriers of
g_i^\beta (to show that the sum is finite). So suppose
you have a point on the boundary of some V_\beta.
What guarantees that it has a neighborhood that
intersects at most finitely many of the carriers of
g_i^\beta? It cannot be the local finiteness of the
partition of unity transported from the manifold
chart domain to V_\beta since that is only locally
finite with respect to the open set V_\beta. I had
the same problem with Lang's proof that I referenced
in my other post (Incorrect proof in Lang's Fundamentals
of Differential Geometry).

Help, again, please.

Thanks,

Jeff- Hide quoted text -

- Show quoted text -

I am not sure I see the problem.Your origional cover of E was refined
to be locally finite so for each point of E there should be a finite
number of tranlates of the open chart domains intersecting the point
and then for each such chart a ball intersecting only a finite number
of supports of the
functions so on the intersection of these balls (open) only a finite
number of functions are non zero. So the sum of a finite number of C^k
functions is C^k .What am I missing ?

I thought of something else .Isn't an open ball in a Banach Space
diffeomorphic to the whole Banach space ? That would simplify the
whole problem.smn
.

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  • Re: Partitions of unity proof roadblock
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  • Re: Partitions of unity proof roadblock
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