Sergei Yakovenko's blog: on Math and Teaching

Monday, February 1, 2016

Finally, exam!

Filed under: lecture,Rothschild course "Analysis for high school teachers" — Sergei Yakovenko @ 3:41
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Exam

The exam is posted online on Feb 1, 2016, and must be submitted on the last day of the exams’ period, February 26. Its goals are, besides testing your acquired skills in the Analysis, to teach you a few extra things and see your ability for logical reasoning, not your proficiency in performing long computations. If you find yourself involved in heavy computations, better double check whether you understand the formulation of the problem correctly. Remember, small details sometimes matter!

Please provide argumentation, better in the form of logical formulas, not forgetting explicit or implicit quantifiers \forall and \exists. They really may change the meaning of what you write!

Problems are often subdivided into items. The order of these items is not accidental, try to solve them from the first till the last, and not in a random order (solution of one item may be a building block for the next one).

To get the maximal grade, it is not necessary to solve all problems, but it is imperative not to write stupid things. Please don’t try to shoot in the air.

The English version is the authoritative source, but if somebody translates it into Hebrew (for the sake of the rest of you) and send me the translation, I will post it for your convenience, but responsibility will be largely with the translator.

If you believe you found an error or crucial omission in the formulation of a problem, please write me. If this will be indeed the case (errare humanum est), the problem will be either edited (in case of minor omissions) or cancelled (on my account).

That’s all, folks!© Good luck to everybody!

Yes, and feel free to leave your questions/talkbacks here, whether addressed to Michal/Boaz/me or to yourself, if you feel you want to ask a relevant question.

Corrections


Correction 1


The formulation of Problem 1 was indeed incorrect. The set A' was intended to be the set of accumulation points for a set A\subseteq [0,1]. The formal definition is as follows.

Definition. A point p\in [0,1] belongs to to the set of limit points A' if and only if \forall\varepsilon>0 the intersection (p-\varepsilon,p+\varepsilon)\cap A is infinite. The point p itself may be or may be not in A.

Isolated points of A are never in A', but A' may contain points p\notin A.

Apologies for the hasty formulation.

Tuesday, January 26, 2016

Lecture 13, Jan 26, 2016

Questions and answers

Questions concerned integrability of discontinuous functions, notions of improper integrals (how and when they can be defined), topological properties (equivalent definitions of compactness, connectedness etc.)

Here are some textbooks that I recommend for preparing when working on the exam. Keep them on your virtual bookshelf: they cover much more that I explained in the course, but who knows what questions related to analysis you might have.

  1. V. Zorich, vol. 1: Chapters 1-6, pp.1-371.
  2. V. Zorich, vol. 2: Parts of Chapter 9 (continuous maps) and the first part of Chapter 18 on Fourier series.
  3. W. Rudin: Chapters 1-6, pp.1-165.

The problems for exam will be posted on February 1st (at least the English version).

Lecture 12, Jan 21, 2016

Filed under: lecture,Rothschild Course "Symmetry" — Sergei Yakovenko @ 4:51
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Basics of the Fourier series

  1. Low regularity functions: example, discussion. Stock market, home electricity meter…
  2. Euclidean geometry in the n-space, real and complex. Normed bases and their advantage. Projections.
  3. Scalar product on the space of functions as the limit of finite-dimensional scalar products.
  4. Fourier coefficients in the real and complex form. Rate of their decay for functions of different regularity.
  5. Fourier series and their main properties (without proofs). Completeness of the exponential/trigonometric system, Parseval equality.
  6. Connection with holomorphic functions. Converging power series as a source of fast converging Fourier series.

Monday, January 18, 2016

Lecture 11, Jan 5, 2016

Functions of complex variable

The field \mathbb C is naturally extending the field \mathbb R, which means that all arithmetic operations on \mathbb R extend naturally as operations on \mathbb C. In particular, any polynomial p(x)=a_0x^n+a_1 x^{n-1}+\cdots+a_n can be interpreted as a map p:\mathbb C\to\mathbb C. Geometrically, this can be visualized as a map of the 2-plane to the 2-plane.

We discussed the maps p(x)=x^2 and p(x)=1/x.

Smooth functions are those which can be accurately approximated by the \mathbb R-affine maps x\mapsto c+\lambda(x-a) near each point a\in\mathbb R. Functions that can be accurately approximated by \mathbb C-affine maps (the same formula, but over the complex numbers), are called holomorphic (or complex analytic). Such maps are characterized by the property that small circles are mapped into small almost-circles, that is,

  1. angles are preserved, and
  2. lengths are scaled

Sometimes these local conditions become global. Examples: the affine maps x\mapsto \lambda (x-a)+b send lines to lines and circles to circles. The map x\mapsto 1/x maps circles and lines into circles or lines (depending on whether the circles/lines pass through the origin x=0).

Complex integration

Integration is carried over smooth (or piecewise smooth) paths in \mathbb C, using Riemann-like sums. It depends linearly on the function which we integrate, but in contrast with the real case we have much more freedom in choosing the paths.

  1. If f(x)=c is a constant, and \gamma =[p_0,p_1]+[p_1,p_2]+[p_2,p_0] is a closed triangle, then the integral is zero as the sum c(p_1-p_0)+c(p_2-p_1)+c(p_0-p_2)=c\cdot0=0.
  2. If f(x) is non-constant, then the integral identity is valid only for “very small triangles” near a point a\in\mathbb C with c=f(a).
  3. This implies that \displaystyle \int_\gamma f(x)\,\mathrm dx=\int_{\gamma'} f(x)\,\mathrm dx as long as the paths \gamma,\gamma' share the common endpoints and can be continuously deformed one into the other.

Integrals over closed loops are zeros, unless there are singular points (where the function is non-holomoprhic) inside.

Example: f(x)=\frac 1x is non-holomorphic at x=0, and \displaystyle \oint_{|x|=1}\tfrac 1x\,\mathrm dx=2\pi i.

Cauchy integral formula

If f is holomorphic inside a domain U bounded by a closed curve \gamma and a\in U, then \displaystyle f(a)=\frac 1{2\pi i}\oint_\gamma \frac{f(x)\,\mathrm dx}{x-a}.

In other words, the value of f on the boundary uniquely determine its values inside the domain. This is in wild contrast with functions of real variable!

Taylor series

As a function of a, the expression \frac1{x-a} admits a converging Taylor expansion (in fact, the same old geometric progression series) in powers of a-a_0 for any a_0\ne x. Thus if we choose a_0\in U off the path \gamma, then the series will converge for any x\in\gamma (warning! note that the “variable” and the “parameter” exchanged their roles!!!), the Cauchy integral can be expanded in the converging series of powers (a-a_0)^n, \ n=0,1,2,\dots, hence the function f(x) gets expanded in the series f(a)=c_0+c_1(a-a_0)+c_2(a-a_0)^2+\cdots.

Conclusion: functions that are \mathbb C-differentiable can be expanded in the convergent Taylor series (hence are “polynomials of infinite degree”) and vice versa, “polynomials of infinite degree” are infinitely \mathbb C-differentiable. This is a miracle that so many functions around us are actually holomorphic!

Tuesday, December 29, 2015

Lecture 10, Dec 29, 2015

Elementary transcendental functions as solutions to simple differential equations

The way how logarithmic, exponential and trigonometric functions are usually introduced, is not very satisfactory and appears artificial. For instance, the mere definition of the non-integer power x^a, a\notin\mathbb Z, is problematic. For a=1/n,\ n\in\mathbb N, one can define the value as the root \sqrt[n]x, but the choice of branch/sign and the possibility of defining it for negative x is speculative. For instance, the functions x^{\frac12} and x^{\frac 24} may turn out to be different, depending on whether the latter is defined as \sqrt[4]{x^2} (makes sense for negative x) or as (\sqrt[4]x)^2 which makes sense only for positive x. But even if we agree that the domain of x^a should be restricted to positive arguments only, still there is a big question why for two close values a=\frac12 and a=\frac{499}{1000} the values, say, \sqrt 2 and \sqrt[1000]{2^{499}} should also be close…

The right way to introduce these functions is by looking at the differential equations which they satisfy.

A differential equation (of the first order) is a relation, usually rational, involving the unknown function y(x), its derivative y'(x) and some known rational functions of the independent variable x. If the relation involves higher derivatives, we say about higher order differential equations. One can also consider systems of differential equations, involving several relations between several unknown functions and their derivatives.

Example. Any relation of the form P(x, y)=0 implicitly defines y as a function of x and can be considered as a trivial equation of order zero.

Example. The equation y'=f(x) with a known function f is a very simple differential equation. If f is integrable (say, continuous), then its solution is given by the integral with variable upper limit, \displaystyle y(x)=\int_p^x f(t)\,\mathrm dt for any meaningful choice of the lower limit p. Any two solutions differ by a constant.

Example. The equation y'=a(x)y with a known function a(x). Even the case where a(x)=a is a constant, there is no, say, polynomial solution to this equation (why?), except for the trivial one y(x)\equiv0. This equation is linear: together with any two functions y_1(x),y_2(x) and any constant \lambda, the functions \lambda y_1(x) and y_1(x)\pm y_2(x) are also solutions.

Example. The equation y'=y^2 has a family of solutions \displaystyle y(x)=-\frac1{x-c} for any choice of the constant c\in\mathbb R (check it!). However, any such solution “explodes” at the point x=c, while the equation itself has no special “misbehavior” at this point (in fact, the equation does not depend on x at all).

Logarithm

The transcendental function y(x)=\ln x satisfies the differential equation y'=x^{-1}: this is the only case of the equation y'=x^n,\ n\in\mathbb Z, which has no rational solution. In fact, all properties of the logarithm follow from the fact that it satisfies the above equation and the constant of integration is chosen so that y(1)=0. In other words, we show that the function defined as the integral \displaystyle \ell(x)=\int_1^x \frac1t\,\mathrm dt possesses all what we want. We show that:

  1. \ell(x) is defined for all x>0, is monotone growing from -\infty to +\infty as x varies from 0 to +\infty.
  2. \ell(x) is infinitely differentiable, concave.
  3. \ell transforms the operation of multiplication (of positive numbers) into the addition: \ell(\lambda x)=\ell(\lambda)+\ell(x) for any x,\lambda>0.

Exponent

The above listed properties of the logarithm ensure that there is an inverse function, denoted provisionally by E(x), which is inverse to \ell:\ \ell(E(x))=x. This function is defined for all real x\in\mathbb R, takes positive values and transforms the addition to the multiplication: E(\lambda+x)=E(\lambda)\cdot E(x). Denoting the the value E(1) by e, we conclude that E(n)=e^n for all n\in\mathbb Z, and E(x)=e^x for all rational values x=\frac pq. Thus the function E(x), defined as the inverse to \ell, gives interpolation of the exponent for all real arguments. A simple calculation shows that E(x) satisfies the differential equation y'=y with the initial condition y(0)=1.

Computation

Consider the integral operator \Phi which sends any (continuous) function f:\mathbb R\to\mathbb R to the function g=\Phi(f) defined by the formula \displaystyle g(x)=f(0)+\int_0^x f(t)\,\mathrm dt. Applying this operator to the function E(x) and using the differential equation, we see that E is a “fixed point” of the transformation \Phi: \Phi(E)+E. This suggests using the following approach to compute the function E: choose a function f_0 and build the sequence of functions f_n=\Phi(f_{n-1}), n=1,2,3,4,\dots. If there exists a limit f_*=\lim f_{n+1}=\lim \Phi(f_n)=\Phi(f_*), then this limit is a fixed point for \Phi.

Note that the action of $\Phi$ can be very easily calculated on the monomials: \displaystyle \Phi\biggl(\frac{x^k}{k!}\biggr)=\frac{x^{k+1}}{(k+1)!} (check it!). Therefore if we start with f_0(x)=1, we obtain the functions $\latex f_n=1+x+\frac12 x^2+\cdots+\frac1{n!}x^n$. This sequence converges to the sum of the infinite series \displaystyle\sum_{n=0}^\infty\frac1{n!}x^n which represents the solution E(x) on the entire real line (check that). This series can be used for a fast approximate calculation of the number e=E(1)=\sum_0^\infty \frac1{n!}.

Differential equations in the complex domain

The function E(ix)=e^{ix} satisfies the differential equation y'=\mathrm iy. The corresponding “motion on the complex plane”, x\mapsto e^{\mathrm ix}, is rotation along the (unit) circle with the unit (absolute) speed, hence the real and imaginary parts of e^{\mathrm ix} are cosine and sine respectively. In fact, the “right” definition of them is exactly like that,

\displaystyle \cos x=\textrm{Re}\,e^{\mathrm ix},\quad \sin x=\textrm{Im}\,e^{\mathrm ix} \iff e^{\mathrm ix}=\cos x+\mathrm i\sin x,\qquad x\in\mathbb R.

Thus, the Euler formula “cis” in fact is the definition of sine and cosine. Of course, it can be “proved” by substituting the imaginary value into the Taylor series for the exponent, collecting the real and imaginary parts and comparing them with the Taylor series for the sine and cosine.

In fact, both sine and cosine are in turn solutions of the real differential equations: derivating the equation y'=\mathrm iy, one concludes that y''=\mathrm i^2y=-y. It can be used to calculate the Taylor coefficients for sine and cosine.

For more details see the lecture notes.


Not completely covered in the class: solution of linear equations with constant coefficients and resonances.

Sunday, December 27, 2015

Lecture 9, Dec 22, 2015

Integral and antiderivative

  1. Area under the graph as a paradigm
  2. Definitions (upper and lower sums, integrability).
  3. Integrability of continuous functions.
  4. Newton-Leibniz formula: integral and antiderivative.
  5. Elementary rules of antiderivation (linearity, anti-Leibniz rule of “integration by parts”).
  6. Anti-chain rule, change of variables in the integral and its geometric meaning.
  7. Riemann–Stieltjes integral and change of variables in it.
  8. Integrability of discontinuous functions.

Not covered in the class: Lebesgue theorem and motivations for transition from Riemann to the Lebesgue integral.

The sketchy notes are available here.

Saturday, December 19, 2015

Lecture 8, Dec 15

Higher derivatives and better approximation

We discussed a few issues:

  • Lagrange interpolation formula: how to estimate the difference f(b)-f(a) through the derivative f'?
  • Consequence: vanishing of several derivatives at a point means that a function has a “root of high order” at this point (with explanation, what does that mean).
  • Taylor formula for polynomials: if you know all derivatives of a polynomial at some point, then you know it everywhere.
  • Peano formula for C^n-smooth functions: approximation by the Taylor polynomial with asymptotic bound for the error.
  • Lagrange formula: explicit estimate for the error.

The notes (updated) are available here.

Saturday, December 12, 2015

Lecture 7, Dec 8, 2015

Differentiability and derivative

Continuity of functions (and maps) means that they can be nicely approximated by constant functions (maps) in a sufficiently small neighborhood of each point. Yet the constant maps (easy to understand as they are) are not the only “simple” maps.

Linear maps

Linear maps naturally live on vector spaces, sets equipped with a special structure. Recall that \mathbb R is algebraically a field: real numbers cane be added, subtracted between themselves and the ratio \alpha/\beta is well defined for \beta\ne0.

Definition. A set V is said to be a vector space (over \mathbb R), if the operations of addition/subtraction V\owns u,v\mapsto u\pm v and multiplication by constant V\owns v,\ \mathbb R\owns \alpha\mapsto \alpha v are defined on it and obey the obvious rules of commutativity, associativity and distributivity. Some people prefer to call vector spaces linear spaces: the two terms are identical.

Warning. There is no “natural” multiplication V\times V\to V!

Examples.

  1. The field \mathbb R itself. If we want to stress that it is considered as a vector space, we write \mathbb R^1.
  2. The set of tuples \mathbb R^n=(x_1,\dots,x_n),\ x_i\in\mathbb R is the Euclidean n-space. For n=2,3 it can be identified with the “geometric” plane and space, using coordinates.
  3. The set of all polynomials of bounded degree \leq d with real coefficients.
  4. The set of all polynomials \mathbb R[x] without any control over the degree.
  5. The set C([0,1]) of all continuous functions on the segment [0,1].

Warning. The two last examples are special: the corresponding spaces are not finite-dimensional (we did not have time to discuss what is the dimension of a linear space in general…)

Let V,Z be two (different or identical) vector spaces and f:V\to W is a function (map) between them.
Definition. The map $f$ is linear, if it preserves the operations on vectors, i.e., \forall v,w\in V,\ \alpha\in\mathbb R,\quad f(v+w)=f(v)+f(w),\ f(\alpha v)=\alpha f(v).

Sometimes we will use the notation V\overset f\longrightarrow Z.

Obvious properties of linearity.

  • f(0)=0 (Note: the two zeros may lie in different spaces!)
  • For any two given spaces V,W the linear maps between them can be added and multiplied by constants in a natural way! If V\overset {f,g}\longrightarrow W, then we define (f+g)(v)=f(v)+g(v) for any v\in V (define \alpha f yourselves). The result will be again a linear map between the same spaces.
  • If V\overset f\longrightarrow W and W\overset g\longrightarrow Z, then the composition g\circ f:V\overset f\longrightarrow W\overset g\longrightarrow Z is well defined and again linear.

Examples.

  1. Any linear map \mathbb R^1\overset f\longrightarrow \mathbb R^1 has the form x\mapsto ax, \ a\in\mathbb R (do you understand why the notations \mathbb R, \mathbb R^1 are used?)
  2. Any linear map \mathbb R^n\overset f\longrightarrow \mathbb R^1 has the form (x_1,\dots,x_n)\mapsto a_1x_1+\cdots+a_nx_n for some numbers a_1,\dots,a_n. Argue that all such maps form a linear space isomorphic to \mathbb R^n back again.
  3. Explain how linear maps from \mathbb R^n to \mathbb R^m can be recorded using n\times m-matrices. How the composition of linear maps is related to the multiplication of matrices?

The first example shows that linear maps of \mathbb R^1 to itself are “labeled” by real numbers (“multiplicators“). Composition of linear maps corresponds to multiplication of the corresponding multiplicators (whence the name). A linear 1-dim map is invertible if and only if the multiplicator is nonzero.

Corollary. Invertible linear maps \mathbb R^1\to\mathbb R^1 constitute a commutative group (by composition) isomorphic to the multiplicative group \mathbb R^*=\mathbb R\smallsetminus \{0\}.

Shifts

Maps of the form V\to V, \ v\mapsto v+h for a fixed vector h\in V (the domain and source coincide!) are called shifts (a.k.a. translations). Warning: The shifts are not linear unless h=0! Composition of two shifts is again a shift.

Exercise.
Prove that all translations form a commutative group (by composition) isomorphic to the space V itself. (Hint: this is a tautological statement).

Affine maps

Definition.
A map f:V\to W between two vector spaces is called affine, if it is a composition of a linear map and translations.

Example.
Any affine map \mathbb R^1\to\mathbb R^1 has the form x\mapsto ax+b for some a,b\in\mathbb R. Sometimes it is more convenient to write the map under the form x\mapsto a(x-c)+b: this is possible for any point c\in\mathbb R^1. Note that the composition of affine maps in dimension 1 is not commutative anymore.

Key computation. Assume you are given a map f:\mathbb R^1\to\mathbb R^1 in the sense that you can evaluate it at any point c\in\mathbb R^1. Suppose an oracle tells you that this map is affine. How can you restore the explicit formula f(x)=a(x-c)+b for f?

Obviously, b=f(c). To find \displaystyle a=\frac{f(x)-b}{x-c}, we have to plug into it any point x\ne c and the corresponding value f(x). Given that b=f(c), we have \displaystyle a=\frac{f(x)-f(c)}{x-c} for any choice of x\ne c.

The expression a_c(x)=\displaystyle \frac{f(x)-f(c)}{x-c} for a non-affine function f is in general not-constant and depends on the choice of the point x.

Definition. A function f:\mathbb R^1\to\mathbb R^1 is called differentiable at the point c, if the above expression for a_c(x), albeit non-constant, has a limit as x\to c:\ a_c(x)=a+s_c(x), where s_c(x) is a function which tends to zero. The number a is called the derivative of f at the point c and denoted by f'(c) (and also by half a dozen of other symbols: \frac{df}{dx}(c),Df(c), D_xf(c), f_x(c), …).

Existence of the limit means that near the point c the function f admits a reasonable approximation by an affine function \ell(x)=a(x-c)+b: f(x)=\ell(x)+s_c(x)(x-c), i.e., the “non-affine part” s_c(x)\cdot (x-c) is small not just by itself, but also relative to small difference x-c.

Differentiability and algebraic operations

See the notes and their earlier version.

The only non-obvious moment is differentiability of the product: the product (unlike the composition) of affine functions is not affine anymore, but is immediately differentiable:

[b+a(x-c)]\cdot[q+p(x-c)]=pq+(aq+bp)(x-c)+ap(x-c)^2, but the quadratic term is vanishing relative to x-c, so the entire sum is differentiable.

Exercise. Derive the Leibniz rule for the derivative of the product.

Derivative and the local study of functions

Affine functions have no (strong) maxima or minima, unless restricted on finite segments. Yet absence of the extremum is a strong property which descends from the affine approximation to the original function. Details here and here.

Sunday, December 6, 2015

Lecture 6, Dec 1, 2015

Two properties preserved by continuous maps: compactness and connectivity

These two properties are key to existence of solutions to infinitely many problems in mathematics and physics.

Compactness

Compactness (of a subset A\subseteq \mathbb R^n) is the “nearest approximation” to finiteness of A. Obviously, if A is a finite set of points, then

  1. Any infinite sequence \{a_n\}_{n=1}^\infty\subseteq A has an infinite stationary (constant) subsequence;
  2. A is bounded and closed;
  3. If \bigcup_\alpha U_\alpha\supseteq A is an arbitrary covering of A by open subsets U_\alpha, then one can always choose a finite subcovering U_{\alpha_1}\cup\cdots\cup U_{\alpha_N}\supseteq A.

The first two properties are obvious, the third one also. For each point a_1,\dots,a_N\in A it is enough to find just one open set U_{\alpha_i} which covers this point. Their union (automatically finite) covers all of A.

Definition.The following three properties of a set A\subseteq \mathbb R^n are equivalent:

  1. Any infinite sequence \{a_n\}_{n=1}^\infty\subseteq A has a partial limit (i.e., the limit of an infinite subsequence), which is again in A;
  2. A is bounded and closed;
  3. If \bigcup_\alpha U_\alpha\supseteq A is an arbitrary covering of A by open subsets U_\alpha, then one can always choose a finite subcovering U_{\alpha_1}\cup\cdots\cup U_{\alpha_N}\supseteq A.

Example. The closed segment, say, [0,1]\subset\mathbb R^1 possesses all three properties.

  1. The standard trick of division into halves and choosing each time the half that contains infinitely many members of the sequence allows to construct a partial limit for any sequence confined to [0,1].
  2. Obvious.
  3. Assume (by contradiction) that there exists a very perverse covering of [0,1], which does not allow for a choice of finite subcovering. Then at least one of the two segments, [0,\frac12],\ [\frac12,1], also suffers from the same problem (if both admit finite subcovering, one would easily construct a finite subcovering for the initial segment [0,1]). Continuing this way, we construct an infinite nested sequence of closed intervals which do not admit a finite subcovering. Their intersection is a point a\in[0,1] which must be covered by at least one open set. But then this set covers also all sufficiently small segments from our nested sequence. Contradiction.

Problem. Prove (using the Example) that the three conditions are indeed equivalent. Hint: any bounded set can be confined to a cube x_i\in [-C_i,C_i],\ i=1,\dots, n. Use the closedness of A to prove that the partial limit of any sequence is again in A.

Theorem. If f\colon A\to \mathbb R^m is a continuous map and A is compact, than f(A) is also compact.

Corollary. Any continuous function restricted on a compact is bounded and attains its extremal values.

Connectivity

A subset A\subseteq \mathbb R^n is called connected, if it cannot be split into two disjoint parts “apart from each other”. How this can be formalized?

Example (proto-Definition). A subset A\subseteq [0,1] is called connected, if together with any two points a,b\in A it contains all points x such that a\le x\le b.

All connected subsets of the real line can be easily described (Do it!).

How can we treat subsets A\subseteq \mathbb R^n for n>1? Two ways can be suggested.

Definition. A set A\subseteq \mathbb R^n is called path connected, if for any two points a,b\in A there exists a continuous map f\colon [0,1]\to A such that f(0)=a,\ f(1)=b.

This definition mimics the one-dimensional construction. However, this is not the only possibility to say that a set cannot be split into smaller parts.

Definition. A subset A\subseteq \mathbb R^n is called disconnected, if there exist two open disjoint sets U_1,U_2\subseteq\mathbb R^n, \ U_1\cap U_2=\varnothing, such that the two parts A\cap U_i, \ i=1,2 are both nonempty. If such partition is impossible, then A is called connected.

Problem. Prove that for subsets on the real line the two definitions coincide.

Problem. Consider the subset of the plane A which consists of the graph y=\sin \frac1x,\ x>0 and the point (0,0). Prove that it is connected but not path connected.

Further reading

Chapter 3 from Abbot, Understanding Analysis. Especially sections 3.2 (open/closed sets), 3.3 (compact sets) and 3.4 (connected sets). Pay attention to the exercises!

Tuesday, November 24, 2015

Lecture 5, Nov 24, 2015

The Peano curve: continuity can be counter-intuitive

The Peano curve is obtained as the limit of piecewise-linear continuous (even closed) curves \gamma_n. Denote by K=\{|x|+|y|\le 1\} the square (rotated by \frac \pi/4) and by \mathbb Z^2=\{(x,y):x,y\in\mathbb Z the grid of horizontal and vertical lines at distance 1 from each other, then one can construct a family of piecewise-linear continuous curves \gamma_n:[0,1]\to\mathbb R^2 which visits all points of the intersection K\cap\frac1{2^n}\mathbb Z^2 in such a way that |\gamma_n(t)-\gamma_n(t)|<\frac1{2^n} uniformly on t\in[0,1].

This sequence of curves converges uniformly to a function (curve) \gamma_*:[0,1]\to\mathbb R^2 and this curve is closed and continuous for the same reasons that justify continuity of the Koch snowflake curve.

What are the properties of the images C_n=\gamma_n([0,1]) and of the limit curve C_*=\gamma_*([0,1])?

  • Each curve C_n for any finite n is piecewise-linear. It has zero area in the sense that for any \varepsilon > 0 the curve C_n can be covered by a finite union of (open) rectangles with the total area less than \varepsilon;
  • Each curve C_n has finite length (although it grows to infinity as n\to\infty, – check it!).
  • The limit curve C_* has no length (that’s the same as saying that it has infinite length). Moreover, unlike many other curves of infinite length (say, the straight line \{y=0\}\subseteq\mathbb R^2), no part \gamma([a,b]),\ a<b, of C_* has finite length!
  • The limit curve C_* coincides with the square K, hence fills the area equal to 2.

All these assertions are easy except for the last one. Let’s prove it.

Consider the images C_n=K\cap \frac1{2^n}\mathbb Z^2. The union of these images is dense in K: by definition, this means that any point P\in K can be approximated by a sequence of points P_n\in C_n which converge to P as n\to\infty. Being in the image of \gamma_n([0,1]), each point P_n is the image of some point in [0,1]: \exists a_n\in[0,1]:\ \gamma(a_n)=P_n. Such point may well be non-unique, and in any case we have absolutely no knowledge of how the points a_1,a_2,\dots are distributed over [0,1].

However, we know that the sequence a_n\in [0,1] must have an accumulation point a_*\in [0,1], which is by definition a limit of some infinite subsequence. (This won’t be the case if instead of [0,1] we were dealing with the curves defined on the entire real line!). Replacing the sequence by this subsequence, we see that it still converges to the same limit, P_n=\gamma(a_n)\to a_*=\gamma_*(a_*)=P. Thus we proved that an arbitrary point in K lies in the image: P\in C_*.

Topology: the study of properties preserved by continuous maps (functions, applications, …)

Definition. A neighborhood of a point a\in\mathbb R^n in the Euclidean space is any set of the form \{x:|x-a| 0, where | ??? | is a distance function satisfying the triangle inequality. Examples:

  • |x|=\sqrt{x_1^2+\cdots+x_n^2} (the usual Euclidean distance on the line, on the plane, …) for x=(x_1,\dots,x_n)\in\mathbb R^n;
  • |x|=\max\{|x_1|, \dots, |x_n|\} (in the above notation);
  • |x|=|x_1|+\cdots+|x_n|.

Definition. A subset A\subset\mathbb R^n of the Euclidean space (OK, plane) is called open, if together with any its point a\in A it contains some neighborhood of a.
A subset is called closed, if the limit of converging infinite sequence \{a_n\}\subset A again belongs in A.

Theorem. A subset A is open if and only if its complement \mathbb R^n\smallsetminus A is closed.

Theorem. The union of any family (infinite or even uncountable) of open sets is open. Finite intersection of open sets is also open (for infinite intersections this is wrong).
Corollary. Intersection of any family (infinite or even uncountable) of closed sets is closed. Finite union of closed sets is also closed (for infinite intersections this is wrong).

One can immediately produce a lot of examples of open/closed subsets in \mathbb R^n. It turns out that any property that can be formulated using only these notions, is preserved by maps which are continuous together with their inverses. The corresponding area of math is called topology.

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