Gauge (Minkowski functional) of Convex Set

Theorem: Let $K$ be a convex set.

(i) $p_K(x)\leq 1$ if $x\in K$.

(ii) $p_K(x)<1$ if and only if $x$ is an interior point of $K$.

We need the following definition: $p_K(x)=\inf\{a\mid a>0,x/a\in K\}$. This $p_K$ is often known as a Gauge or Minkowski functional.

Proof:

(i) If $x\in K$, then $x/1\in K$, so $P_K(x)\leq 1$. This is the easy part. The converse holds here, if $x\notin K$, then $p_K(x)>1$.

(ii) We first prove the “only if” part. Assume $p_K(x)<1$. Suppose $x$ is not an interior point of $K$, i.e. there exists $y\in X$ such that for all $\epsilon>0$, $x+ty\notin K$ for some $|t|<\epsilon$.

We then have $p_K(x+ty)>1$. Combining with the subadditive property of the gauge, we have $1. Rearranging, we get $p_K(x)>1-p_K(ty)$. By considering the various possibilities of the sign of $t$, and using the positive homogeneity of the gauge, we can obtain a contradiction. For example, if $t>0$, $p_K(x)>1-tp_K(y)$. Since $t\to 0$ as $\epsilon\to 0$, this implies $p_K(x)\geq 1$, a contradiction.

Conversely, if $x$ is an interior point of $K$, for all $y$ there exists $\epsilon>0$ such that $x+ty\in K$ for all $|t|<\epsilon$.

We have $p_K(x+ty)\leq 1$ for all $y$, for all $|t|<\epsilon$. Since it is a “for all” quantifier, we can choose in particular $y=x$, $t>0$.

Then we have $p_K((1+t)x)\leq 1$, which leads to $(1+t)p_K(x)\leq 1$ and $p_K(x)\leq \frac{1}{1+t}<1$.

Recommended Functional Analysis Book (Graduate Level)

Functional Analysis is a subject that combines Linear Algebra with Analysis. I researched online, and it seems one of the best Functional Analysis Book for Graduate level is Functional Analysis by Peter Lax. This book is ideal for a second course in Functional Analysis. For a first course in Functional Analysis, I would recommend Kreyszig, which is listed on my Recommended Undergraduate Math Books page.

This is Theorem 5 in the book: Let $X$ be a linear space over the reals.

The following 9 properties hold:

1. The empty set is convex.
2. A subset consisting of a single point is convex.
3. Every linear subspace of $X$ is convex.
4. The sum of two convex subsets is convex.
5. If $K$ is convex, so is $-K$.
6. The intersection of an arbitrary collection of convex sets is convex.
7. Let $\{K_j\}$ be a collection of convex subsets that is totally ordered by inclusion. Then their union $\cup K_j$ is convex.
8. The image of a convex set under a linear map is convex.
9. The inverse image of a convex set under a linear map is convex.

Brief sketch of proofs:

We give a brief sketch of the idea behind the proofs.

We are using the definition of convex as follows: $X$ is a linear space over the reals; a subset $K$ of $X$ is called convex if, whenever $x$ and $y$ belong to $K$, all points of the form $ax+(1-a)y$, $0\leq a\leq 1$ also belong to $K$.

Property 1 is vacuously true.

Property 2 is true because of $ax+(1-a)x\equiv x$.

Property 3 is true because $ax+(1-a)y$ is a linear combination and is thus in the linear subspace.

Property 4) Let $C_1$ and $C_2$ be the two convex subsets. Let $x_1+y_1$ and $x_2+y_2$ be points in $C_1+C_2=\{x+y:x\in C_1, y\in C_2\}$

\begin{aligned} a(x_1+y_1)+(1-a)(x_2+y_2)&=ax_1+ay_1+x_2+y_2-ax_2-ay_2\\ &=[ax_1+(1-a)x_2]+[ay_1+(1-a)y_2]\\ &\in C_1+C_2 \end{aligned}

Property 5) We just need to know that $-K=\{-x:x\in K\}$ and this algebraic observation: $a(-x)+(1-a)(-y)=-(ax+(1-a)y)$.

Property 6) Let $x,y\in\bigcap_{i\in I}C_i$. $ax+(1-a)y\in C_i$ for all $i\in I$, thus $ax+(1-a)y\in\bigcap_{i\in I}C_i$.

Property 7) Let $x,y\in\bigcup K_j$, where $K_i\subseteq K_{i+1}$. Let $x\in K_n$, $y\in K_m$, then either $K_n\subseteq K_m$ or $K_m\subseteq K_n$. If $K_n\subseteq K_m$, $ax+(1-a)y\subseteq K_m\subseteq \bigcup K_j$. Similarly for the other case $K_m\subseteq K_n$.

Property 8) Observe that $af(x)+(1-a)f(y)=f(ax+(1-a)y)\in f(K)$.

Property 9) The only tricky thing about this part is that we cannot assume that the inverse $f^{-1}$ exists. We can only talk about the pre-image.

Let $w,z\in f^{-1}(K)$. $f(w)\in K$ and $f(z)\in K$.

We have $f(aw+(1-a)z)=af(w)+(1-a)f(z)\in K$.

Thus $aw+(1-a)z\in f^{-1}(K)$.

The End!

Free Coursera Course: An Introduction to Functional Analysis

There is an interesting upcoming course at Coursera, suitable for undergraduates! (Starting 12 September 2014) Join the class if interested, it is free! Functional Analysis is actually a third year course for Math Majors at university. There are some powerful and deep theorems in functional analysis, like the Riesz representation theorem.

Functional analysis is the branch of mathematics dealing with spaces of functions. It is a valuable tool in theoretical mathematics as well as engineering. It is at the very core of numerical simulation.

In this class, I will explain the concepts of convergence and talk about topology. You will understand the difference between strong convergence and weak convergence. You will also see how these two concepts can be used.

You will learn about different types of spaces including metric spaces, Banach Spaces, Hilbert Spaces and what property can be expected. You will see beautiful lemmas and theorems such as Riesz and Lax-Milgram and I will also describe Lp spaces, Sobolev spaces and provide a few details about PDEs, or Partial Differential Equations.

Course Syllabus

Week 1: Topology; continuity and convergence of a sequence in a topological space.
Week 2: Metric and normed spaces; completeness
Week 3: Banach spaces; linear continuous functions; weak topology
Week 4: Hilbert spaces; The Riesz representation theorem
Week 5: The Lax-Milgram Lemma
Week 6: Properties of the Lp spaces
Week 7: Distributions and Sobolev Spaces
Week 8: Application: simulating a membrane

Recommended Background

The course is mostly self-contained; however, you need to be familiar with functions, derivatives and integrals. You need to know what A ∩ B means and to know what a proof is. You should be fine if you have taken Calculus II and Algebra II. Students in Europe who have taken 120 ECTS in science should be fine as well.

Because this is an online class, having advanced and non-advanced students in a class will not be a problem; on the contrary we expect a wide range of interesting interactions. However, non-advanced students may have to work a bit more.

Course Format

The class will consist of a series of lecture videos, usually between five and twelve minutes in length.  There will be approximately one hour worth of video content per week. Some of the videos contain integrated quiz questions. There will also be standalone quizzes that are not part of the video lectures; you will be asked to solve some problems and evaluate the solutions proposed by your fellow classmates. There will be a final exam.

There will be some additional contents in the form of PDF files.

FAQ

• Will I get a Statement of Accomplishment after completing this class?
Yes. Students who successfully complete the class will receive a Statement of Accomplishment signed by the instructor.
• What resources will I need for this class?
For this course, all you need is an Internet connection and the time to view the videos, understand the material, discuss the material with fellow classmates, take the quizzes and solve the problems.
• What pedagogy will be used?
This MOOC is in English but the math will be taught with a “French Touch”.
• What does “teaching math with a French touch” mean?
France has a long-standing tradition where math is addressed from a theoretical standpoint and studied for its implicit value throughout high school and preparatory school for the high-level entrance exams. This leads to a mindset based on proofs and abstraction. This mindset has consequences on problem solving that is sometimes referred to as the “French Engineer”. In contrast, other countries have a tradition where math is addressed as a computation tool.
• Does it mean it will abstract and complicated?
The approach will be rather abstract but I will be sure to emphasize the concepts over the technicalities. Above all, my aim is to help you understand the material and the beauty behind it.

Featured book:

Introductory Functional Analysis with Applications

This is the recommended textbook that covers the material in the Coursera Course (and more).