Weierstrass theorem

According to this theorem, a function /(x) in a closed and bounded domain must attain minimum and maximum values. This theorem is also known as the Extreme Value Theorem. 

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Up to and including Chapter 7, we make very few assumptions in particular, we do not need to know the functional form of the force on the electron. We assume only that this force is spherically symmetric. Yet, armed with some powerful undergraduate-level mathematics (plus Fubini s Theorem and the Stone-Weierstrass Theorem), we can make meaningful predictions from the meager assumptions of the basic model of quantum mechanics and spherical symmetry. 

The goal of this section is to find useful spanning subspaces of C[— 1, 1] and 2(52) Recall from Definition 3.7 that a subspace spans if the perpendicular subspace is trivial. In a finite-dimensional space V, there are no proper spanning subspaces any subspace that spans must have the same dimension as V and hence is equal to V. However, for an infinite-dimensional complex scalar product space the situation is more complicated. There are often proper subspaces that span. We will see that polynomials span both C[—l, 1] andL2(5 2) in Propositions 3.8 and 3.9, respectively. In the process, we will appeal to the Stone-Weierstrass theorem (Theorem 3.2) without giving its proof. 

The Stone-Weierstrass theorem uses another notion of approximation uniform approximation. 

Propositions 3.8 and 3.9 below are both consequences of Proposition 3.7 and the Stone-Weierstrass theorem. Before stating the Stone-Weierstrass theorem, we must define compactness for subsets of R". 

Weierstrass Theorem. Recall the complex scalar product space C[—1,1] of continuous functions on [—1, 1], introduced in Section 2.1. 

This fact will be at the heart of the proof of our main result in Section 6.5. Proof. First, we show that V satisfies the hypotheses of the Stone-Weierstrass theorem. We know that V is a complex vector space under the usual addition and scalar multiplication of functions adding two polynomials or multiplying a polynomial by a constant yields a polynomial. The product of two polynomials is a polynomial. To see that V is closed under complex conjugation, note that for any x e [—1, 1] and any constant complex numbers flo, , a sN[Pg.102]

Proof. We start by showing that V satisfies the hypotheses of the Stone-Weierstrass theorem. Most of the hypotheses follow easily from the fact that polynomials form a vector space closed under multiplication and complex conjugation. It remains to show that the restrictions of polynomials separate points on the two-sphere S. Suppose we have two points, (xi, yi, Z2) and (x2, y2, Z2) such that (xi, yi, Z2) fe, yz, zz)- Then either xi 7 X2 or yi y2 or zi 7 Z2- In the first case, the polynomial x takes different values at the two points. In the second case y does and in the third case z does. So V separates points on the two-sphere S. Hence by Proposition 3.1 we have... 

Stone-Weierstrass theorem will allow us to be sure from our armchairs, with-... 

Exercise 3.22 Show that the set of harmonic polynomials on is not closed under multiplication. (The point of this exercise is that in Chapter 7, when we wish to show that the restrictions of harmonic polyn omials to S " span S, we will not be able to appeal directly to the Stone-Weierstrass theorem. Rather, we will relate restrictions of harmonic functions to restrictions of polynomial functions and then appeal to the results of Section 3.5.)... 

Stone-Weierstrass theorem to conclude that any function f e 1] can... 

Proposition 7.3 implies that any polynomial on the two-sphere in E can be written as a sum of harmonic polynomials. See Exercise 7.3. This fact is important to the proof of Proposition 7.4. The point is that we cannot apply the Stone-Weierstrass theorem directly to harmonic functions (see Exercise 3.22). However, we can apply the Stone-Weierstrass theorem to polynomials. Proposition 7.3 is the link we need. 

The set TR y satisfies the hypotheses of the Stone-Weierstrass theorem. The set Br is compact by Exercise 3.30. The set Tr 0 y is a complex vector space because it is a tensor product of vector spaces. To see that Tr 0 y is closed under multiplication, it suffices to consider products of elements of the form / 0 y we have... 

It follows that the conclusion of the Stone-Weierstrass Theorem holds any continuous function in (Br) can be uniformly approximated by elements of Ir 3. Hence by Proposition 3.7, any element of Br ) can be approximated in the norm by an element of Ir y. 

The results of this section, even with their limitations, are the punch line of our story, the particularly beautiful goal promised in the preface. Now is a perfect time for the reader to take a few moments to reflect on the journey. We have studied a significant amount of mathematics, including approximations in vector spaces of functions, representations, invariance, isomorphism, irreducibility and tensor products. We have used some big theorems, such as the Stone-Weierstrass Theorem, Fubini s Theorem and the Spectral Theorem. Was it worth it And, putting aside any aesthetic pleasure the reader may have experienced, was it worth it from the experimental point of view In other words, are the predictions of this section worth the effort of building the mathematical machinery ... 

This series converges for any stable system within the radius of convergence defined by the input ensemble. If the multivariate function F is not analytic but is continuous, then a finite-order Volterra model can be found that achieves any desirable degree of approximation (based on the Stone-Weierstrass theorem). 

The Bolzano-Weierstrass Theorem (Section 9.17, p. 277) assures that this sequence contains a subsequence that is convergent on some value, say. Wo- As w approaches Wo, approaches t, thereby implying... 

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