From: "Ashok.R"
Subject: More Questions on Functional Analysis.
Date: Fri, 17 Sep 1999 13:53:54 -0400
Newsgroups: sci.math
Keywords: "alignment" in normed linear spaces
Sometime back, I posted some questions on Functional Analysis and a lot
of people gave me some really good explanations. Before I bother the
good people of sci.math once more, I would like to thank Dan Grubb,
Stephen Montgomery Smith and ZVK. The explanations make much more sense
to me now.
Ok ..I am a little bit better than what I started with - although there
are still many many many holes to be patched up.
1. Alignment: I do not have a very good hold on what it is. The
definition goes something like this...Let X be a normed linear space and
let X_Star be the space of all bounded linear functionals that can
operate on elements of X (the dual space of X), then x in X and x_star
in X_star are said to be aligned when
= norm(x)norm(x_star), where is read as " the
value of the functional x_star at x"
I can visualize this in the specific case of the hilbert space when
implies the inner product for fixed second argument and the
inner product is the same as the product of the norms when the two
vectors are linearly dependent (ie. aligned). I have difficulty
visualizing alignment in a general normed space.
For instance, there is a theorem (derrived from the Hahn Banach theorem
actually) that states that "Let x be an element in a real normed linear
space X and let d denote its distance from the subspace M. Then,
d = inf(norm(x-m)) = max() for all x_star having norm <=1 and
for all x_star in M_perp.
Where the maximum on the right is achieved for some x0_star in M_perp.
If the infimum on the left is achieved for some m0 in M then x0_star is
ALIGNED with x-m0. "
M_perp is the space of all functionals that operate on all elements of x
to give zero.
I have no idea what ALIGNMENT means in this context other than the
mathematical definition,
= norm(x-mo)norm(x0_star)
Any suggestions on how to visualize this alignment thingy would help me
a great deal.
2. The Hahn Banach theorem : The way I see it , it says that it is
possible to extend a functional defined in a subspace to a functional
for the whole space, whitout blowing up the norm. The thing that botehrs
me is that The HB theorem is used only in the proof of other thorems and
Luenberger does not give any applications of the theorem as a stand
alone entity. Are there any such applications?
Also Luenbeger claims that the HB theorem is a generalization of the
projection theorem in a Hilbert space. I do not get this at all.
3. (This question may sound really wierd). I am a wannabe controls
engineer and there is this really neat result called the Algebraic
Ricatti Equation (ARE). This tells me how to determine the optimal state
vector for a given quadratic cost function. The opimal state is then
used to get an optimal control law.
Now...I would like to find out if there is any link between the
ARE stuff and the stuff in convex optimization(Minimizing or maximizing
a functional given some convex constraints). Something tells me that
there IS such a connection - although I cannot lay my finger on it.
I suspect that the prrof the ARE uses some functional analysis concepts.
Does anyone know a place where they have good proof for the ARE - that
is like really easy to follow.
Thank you very much. And I hope you were not in Floyd's path - like we
were. Very very wet outside :)
Ashok.R - starving grad student (please feed me!)
http://www.dartmouth.edu/~ashokr/ashokr
==============================================================================
From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik)
Subject: Re: More Questions on Functional Analysis.
Date: 17 Sep 1999 17:05:52 -0400
Newsgroups: sci.math
In article <37E28032.808653F3@dartmouth.edu>,
Ashok.R wrote:
[...]
>1. Alignment: I do not have a very good hold on what it is. The
>definition goes something like this...Let X be a normed linear space and
>let X_Star be the space of all bounded linear functionals that can
>operate on elements of X (the dual space of X), then x in X and x_star
>in X_star are said to be aligned when
>
> = norm(x)norm(x_star), where is read as " the
>value of the functional x_star at x"
>
>I can visualize this in the specific case of the hilbert space when
> implies the inner product for fixed second argument and the
>inner product is the same as the product of the norms when the two
>vectors are linearly dependent (ie. aligned). I have difficulty
>visualizing alignment in a general normed space.
[...]
There are examples in Luenberger what alignment means in concrete spaces.
Let me reduce it to finite dimension:
In the space l_1^3 (3-dimensional, normed by sum of absolute values),
consider the vector
x = [ 2, -3, 4 ]
Its l_1 norm is 9. The question answered by alignment is: Is there a
vector in the dual of l_1 which has norm 1 and "shows" the norm of x?
The dual norm to that od l_1^3 is the norm of l_infinity^3. Let me
pretend that I do not know the aligned vectors to x.
A vector from l_infinity^3 will look like y = [u, v, w], and the norm
inequality says
abs() <= max(|u|, |v|, |w|) * sum(abs(2, -3, 4))
When is the equality attained, provided norm(y)>0?
The absolute values of u, v, w must be equal (say equal to 1), and the
signs must be such that 2*u + (-3)*v + 4*w = 9. (Don't take my word for
it, prove it to yourself!)
This leaves me with
[u, v, w] = [1, -1, 1]
And this [1, -1, 1] is that unit dual vector aligned with [2, -3, 4].
Some vectors have infinitely many unit vectors aligned with them: if
x = [4 0 -7] in l_1^3,
then every vector y = [1 t -1] with -1 <= t <= 1 is aligned with x
(because such a t will not change the dot product, and will keep the
norm at 1.)
Dually, for vectors from l_infinity^n, the dual vectors come from l_1^n,
and trying to align a unit vector y with
x = [3 0 -4 2]
we need to place a (-1) against (-4) and zeros everywhere else:
y = [0 0 -1 0]
Again, non-uniqueness may occur, for example
x = [2 -2 0 0 1]
has aligned unit dual vectors
y = [t t-1 0 0 0] with 0 <= t <= 1 (check it out!).
Geometrically, alignment is an attempt to generalize the "normal vector to
the surface of the unit ball" in case the "ball" has corners. The unit
ball in l_infinity^3 is a cube, and if a primal vector has its endpoint
inside a face, the aligned vector points where we are used to see it
point. If the primal vector's endpoint is on an edge or corner, the
aligned vectors come in a variety of directions, as illustrated by the
examples. (There is another explanation, using supporting hyperplanes,
which are in the primal space, and thir normals are in the dual space.)
Hope it may help, ZVK(Slavek).