In this post we will use the Krein–Milman theorem together with the Hahn–Banach theorem to give another proof of the Stone–Weierstrass theorem. The proof we present does not make use of the classical Weierstrass approximation theorem, so we will have here an alternative proof of the classical theorem as well.
Category Archives: Functional analysis
Advanced Analysis, Notes 13: Banach spaces (convex hulls and the Krein–Milman theorem)
It would be strange to disappear for a week without explanations. This blog was not working for the past week because of the situation in Israel. The dedication in the beginning of the previous post had something to do with this, too. We are now back to work, with the modest hope that things will remain quiet until the end of the semester. We begin our last chapter in basic functional analysis, convexity and the Krein–Milman theorem.
Advanced Analysis, Notes 11: Banach spaces (weak topologies, Alaoglu’s theorem)
Let $latex X$ be the Banach space $latex C([0,1])$ of continuous functions on the interval $latex [0,1]$ with the sup norm. Consider the following sequence of functions $latex {f_n}$ defind as follows. $latex f(0) = 0 $ and $latex f_n(1/(n+1)) = 1$ for all $latex n = 1, 2, ldots$, $latex f_n$ is equal to zero on the interval between $latex 2/(n+1)$ and $latex 1$, and $latex f_n$ is linear in the intervals where we haven’t defined it yet (visualize!). The sequence is tending to zero pointwise, but the norm of $latex X$ does not detect this. The sequence tends to $latex 0$ in the $latex L^1$ norm, but the $latex L^1$ norm is not in the game. Can the Banach space structure of $latex X$ detect the convergence of $latex f_n$ to $latex 0$? Continue reading Advanced Analysis, Notes 11: Banach spaces (weak topologies, Alaoglu’s theorem)
Advanced Analysis, Notes 10: Banach spaces (application: divergence of Fourier series)
Recall Theorem 6 from Notes 3:
Theorem 6: For every $latex f in C_{per}([0,1]) cap C^1([0,1])$, the Fourier series of $latex f$ converges uniformly to $latex f$.
It is natural to ask how much can we weaken the assumptions of the theorem and still have uniform convergence, or how much can we weaken and still have pointwise convergence. Does the Fourier series of a continuous (and periodic) function always converge? In this post we will use the principle of uniform boundedness to see that the answer to this question is a very big NO.
Once again, we begin with some analytical preparations. Continue reading Advanced Analysis, Notes 10: Banach spaces (application: divergence of Fourier series)
Advanced Analysis, Notes 9: Banach spaces (the three big theorems)
Until now we had not yet seen a theorem about Banach spaces — the Hahn–Banach theorems did not require the space to be complete. In this post we learn the three big theorems about operators on Banach spaces: the principle of uniform boundedness, the open mapping theorem, and the closed graph theorem. It is common that these three theorems are presented in texts on functional analysis under the heading “consequences of the Baire category theorem“. Continue reading Advanced Analysis, Notes 9: Banach spaces (the three big theorems)
Advanced Analysis, Notes 8: Banach spaces (application: weak solutions to PDEs)
Today I will show you an application of the Hahn-Banach Theorem to partial differential equations (PDEs). I learned this application in a seminar in functional analysis, run by Haim Brezis, that I was fortunate to attend in the spring of 2008 at the Technion.
As often happens with serious applications of functional analysis, there is some preparatory material to go over, namely, weak solutions to PDEs.
Continue reading Advanced Analysis, Notes 8: Banach spaces (application: weak solutions to PDEs)
Advanced Analysis, Notes 7: Banach spaces (dual spaces and duality, Lp spaces, the double dual, quotient spaces)
Today we continue our treatment of the dual space $latex X^*$ of a normed space (usually Banach) $latex X$. We start by considering a wide class of Banach spaces and their duals. Continue reading Advanced Analysis, Notes 7: Banach spaces (dual spaces and duality, Lp spaces, the double dual, quotient spaces)
Advanced Analysis, Notes 6: Banach spaces (basics, the Hahn-Banach Theorems)
Recall that a norm on a (real or complex) vector space $latex X$ is a function $latex | cdot | : X rightarrow [0, infty)$ that satisfies for all $latex x,y in X$ and all scalars $latex a$ the following:
- $latex |x| = 0 Leftrightarrow x = 0$.
- $latex |ax| = |a| |x|$.
- $latex |x + y | leq |x| + |y|$.
A vector space with a norm on it is said to be a normed space. Inner product spaces are normed spaces. However, many norms of interest are not induced by an inner product. In fact:
Exercise A: A norm is induced by an inner product if and only if it satisfies the parallelogram law:
$latex |x+y|^2 + |x-y|^2 = 2 |x|^2 + 2|y|^2 .$
Instead of solving this exercise, you might prefer to read this old paper where Jordan and von Neumann prove this.
Using Exercise A, it is not hard to show that some very frequently occurring norms, such as the sup norm on $latex C(X)$ or the operator norm on $latex B(H)$, are not induced by inner products. The latter example shows that even if one is working in the setting of Hilbert spaces one is led to study other normed spaces. We now begin our study of normed spaces and, particular, Banach spaces.
Continue reading Advanced Analysis, Notes 6: Banach spaces (basics, the Hahn-Banach Theorems)
Advanced Analysis, Notes 5: Hilbert spaces (application: Von Neumann’s mean ergodic theorem)
In this lecture we give an application of elementary operators-on-Hilbert-space theory, by proving von Neumann’s mean ergodic theorem. See also this treatment by Terry Tao on his blog.
For today’s lecture we will require the following simple fact which I forgot to mention in the previous one.
Exercise A: Let $latex A, B in B(H)$. Then $latex |AB| leq |A| |B|$.
Advanced Analysis, Notes 4: Hilbert spaces (bounded operators, Riesz Theorem, adjoint)
Up to this point we studied Hilbert spaces as they sat there and did nothing. But the central subject in the study of Hilbert spaces is the theory of the operators that act on them. Paul Halmos, in his classic paper “Ten Problem in Hilbert Space“, wrote:
Nobody, except topologists, is interested in problems about Hilbert space; the people who work in Hilbert space are interested in problems about operators.
Of course, Halmos was exaggerating; topologists don’t really care much for Hilbert spaces for their own sake, and functional analysts have much more to say about the structure theory of Hilbert space then what we have learned. Nevertheless, this quote is very close to the truth. We proceed to study operators. Continue reading Advanced Analysis, Notes 4: Hilbert spaces (bounded operators, Riesz Theorem, adjoint)