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).