Lp-TOP dual norm theorem

Theorem: the norm of any sval is the norm of the shortest vector in
the coset of vals mapping to the sval under the V-map sending vals to
svals.

Proof: Say you have V, the vector space of JI intervals represented as
monzos, and G, a subspace of it representing a group of (real)
smonzos. The vector space G* is then the space of svals on these
smonzos. If V is given an Lp norm, then there exists an induced norm
|| · || on G which measures the complexity of smonzos. There then
exists a dual norm || · ||* on G*, which is defined as follows: ||f||*
= sup f(g)/||g|| for f in G* and g in G.

For any subspace G in V, a subspace G° in V* exists representing the
set of all vals tempering out G. The quotient space V*/G° is
isomorphic to the space of svals G*. Using the corollary to the
Hahn-Banach theorem which Claudi just posted
(http://www.math.unl.edu/~s-bbockel1/928/node25.html), we can see that
V*/G° is also isometrically isomorphic to G*, so that the norm on G*
can be recovered from the induced quotient norm on V*/G°.

The quotient norm || · ||~ is given as follows: ||v||~ = inf ||v-g||
for g in G°. In other words, the quotient norm on any coset of V*/G°
is the minimum of the norms of the vectors in the coset. The norm || ·
||* on G* is thus given by ||g||* = ||f+G°||~, where g is in G* and
f+G° is the coset of V*/G° mapping to g under the isometric
isomorphism mentioned above.

Therefore, the norm of any vector v in G* is the same as the minimum
of the norms of the vectors in the coset of V*/G° mapping to v under
the above isometric isomorphism.

-Mike

On Fri, Jul 20, 2012 at 6:28 AM, Mike Battaglia <battaglia01@gmail.com> wrote:
>
> The quotient norm || · ||~ is given as follows: ||v||~ = inf ||v-g||
> for g in G°. In other words, the quotient norm on any coset of V*/G°
> is the minimum of the norms of the vectors in the coset. The norm || ·
> ||* on G* is thus given by ||g||* = ||f+G°||~, where g is in G* and
> f+G° is the coset of V*/G° mapping to g under the isometric
> isomorphism mentioned above.

Now, here's a more interesting question. The quotient norm defined
above sets the norm of any coset in V*/G° as the norm of the shortest
vector in the coset. Now consider the set of all vectors in V* which
is the the shortest vector in its corresponding coset in V*/G°. What
sort of structure does this set have? Will it also end up being a
subspace of V*? If it does, it'd mean that the norm of any sval can
easily be computed by finding the element in the coset of vals mapping
to the sval which lies in this subspace.

I have a strong hunch that it is indeed going to be a subspace of V*,
and that it's going to be the orthogonal complement to G°. IOW, I
conjecture it's going to be the same subspace as G, but translated
into the dual space so that all of the monzos are rewritten as vals.

I'm also aware that if your original Lp space is L1 or Linf, there may
not be a single unique vector in the coset with minimum norm, so this
would have to apply to Lp norms for 0 < p < inf.

-Mike