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From: Joubert on 9 Jul 2010 10:46
In a Sobolev space setting, (H^1)_0 to be precise, my book proves (using
poincare' inequality) that two inner products are equivalent by proving
that the norms they generate are equivalent. Why is this true, in
general? I mean why does the equivalence of two norms imply the
equivalence of the inner products that generate them?
From: Chip Eastham on 9 Jul 2010 15:20
On Jul 9, 10:46 am, Joubert <waterhemlock8...(a)gmail.com> wrote:
> In a Sobolev space setting, (H^1)_0 to be precise, my book proves (using
> poincare' inequality) that two inner products are equivalent by proving
> that the norms they generate are equivalent. Why is this true, in
> general? I mean why does the equivalence of two norms imply the
> equivalence of the inner products that generate them?

In part the answer depends on what you mean by "equivalence"
of two inner products. You might mean that they are equivalent
if they give the same topology of the vector space, and if so
the topologies (metric) depend only on the corresponding norms,
and the proof is evident (the inner products give the same topology
if and only if the respective norms give the same topology).

On the other hand you might have in mind a different definition of
the equivalence of two inner products, say that each is "coercive"
with respect to the norm defined by the other. That sets us up to
do a bit more work. Knowing each is coercive wrt to (the norm of)
the other tells us there are positive constants which bound each norm
with respect to the other, so that direction (implying the norms
are equivalent) works. Going in the other direction is really just
a matter of reversing the argument: if each norm is bounded with
respect to the other, then the inner products are coercive with
respect to the other's norm (by reciprocating a positive constant).

regards, chip




From: David C. Ullrich on 10 Jul 2010 09:29
On Fri, 09 Jul 2010 16:46:18 +0200, Joubert
<waterhemlock8985(a)gmail.com> wrote:

>In a Sobolev space setting, (H^1)_0 to be precise, my book proves (using
>poincare' inequality) that two inner products are equivalent by proving
>that the norms they generate are equivalent. Why is this true, in
>general? I mean why does the equivalence of two norms imply the
>equivalence of the inner products that generate them?

??? What does it _mean_ to say that two inner products
are "equivalent"?




From: Joubert on 10 Jul 2010 11:29

> ??? What does it _mean_ to say that two inner products
> are "equivalent"?


The proof states that a particular functional, continuous w.r.t. to the
usual norm of (H^1)_0, is continuous also w.r.t. the norm defined by
some new scalar product (so as to use Riesz representation theorem with
this new scalar product and show the existence of a weak solution to a
Dirichlet problem). The continuity w.r.t. the new scalar product relies
on the "equivalence" of the scalar products. My guess is two scalar
products are equivalent iff they define equivalent norms, but how does
one show it "by hand"? Furthermore: is my interpretation correct?
From: Joubert on 10 Jul 2010 11:32
On 09/07/2010 21.20, Chip Eastham wrote:

> In part the answer depends on what you mean by "equivalence"
> of two inner products. You might mean that they are equivalent
> if they give the same topology of the vector space, and if so
> the topologies (metric) depend only on the corresponding norms,
> and the proof is evident (the inner products give the same topology
> if and only if the respective norms give the same topology).

> regards, chip

Hmm ok so my interpretation below is right I guess. Thanks.
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