As described in Chapter A, the Euclidean space $\R^d=R\times\cdots\times\R$ is the set of all ordered $d$-tuples or vectors over the real numbers $\R$ (when $d=1$ we refer to the vectors as scalars). /FormType 1 >> Show that (X,d 1) in Example 5 is a metric space. An open interval (0, 1) is an open set in R with its usual metric. 1.2 Examples 1.2.1 The vector space Vof lists The rst example of an in nite dimensional vector space is the space Vof lists of real numbers. Each row corresponds to a different value of $p$. This implies that $A$ is a union of open balls and is therefore an open set. /Type /XObject Show that (X,d 2) in Example 5 is a metric space. There exists some $N$ such that $n>N$ corresponds to $n\epsilon > 1$, implying that the components $x_{N+1}, x_{N+2},\ldots$ of vectors $\bb x\in B_{\epsilon}(\bb 0)$ are unrestricted. 4. Introduction Let X be an arbitrary set, which could consist of vectors in Rn, functions, sequences, matrices, etc. Typical examples are the real numbers or any Euclidean space. $ For example the vector $\bb x=(x_1,\ldots,x_d)$ has $d$ scalar components $x_i\in\R$, $i=1,\ldots,d$. \end{pmatrix}, The basic examples of vector spaces are the Euclidean spaces Rk. Sometimes, we will write d 2 for the Euclidean metric. A metric space X with distance function d is Euclidean if it is (isometrically) embeddable in a Euclidean space. Metric Spaces A metric space is a set X endowed with a metric ρ : X × X → [0,∞) that satisfies ... plane. stream Proposition B.4.5. (M2) d(x, y) = 0 if and only if x = y. Weighted norms are convenient for emphasizing some dimensions over others. \[ d(\bb x,\bb y) = \sqrt{\sum_{i=1}^d (x_i-y_i)^2},\] /ProcSet [ /PDF /Text ] \begin{align*} /Subtype /Form METRIC AND TOPOLOGICAL SPACES 3 1. (8) A space S is semi-locally-connected at a point … Unfortunately, and as usual, it can mean several different things. (we refer to sets of the form (*) as simple cylinders.) PROBLEM 3: Prove that in Euclidean space the boundary of the single point set is itself: @fpg= fpg. It is harder to prove the triangle inequality for the Euclidean metric than For metric spaces (but not for general topological spaces) separability is equivalent to second-countability and also to the Lindelöf property. A metric space M M M is called complete if every Cauchy sequence in M M M converges. NOTES ON METRIC SPACES JUAN PABLO XANDRI 1. Any unbounded subset of any metric space. A union of intersections of a finite number of simple cylinders is a union of open sets and therefore is open also. Euclidean space was introduced by ancient Greeks as an abstraction of our physical space. In fact, whenever we refer to the metric structure of $\R^{\infty}$ we will assume the metric structure of $\bar d$ derived above. /Font << /F38 14 0 R /F15 19 0 R /F44 24 0 R /F21 29 0 R /F40 34 0 R /F18 39 0 R /F35 44 0 R /F41 49 0 R /F42 54 0 R /F39 59 0 R >> /Resources << \def\R{\mathbb{R}} Repeating this for every ${\bb x}\in G$ and taking the union of the resulting rational balls completes the proof. x��YMs�6��W�7jZ��&�Li���Ng�I��ʔ��&ί�. /Filter /FlateDecode Examples of complete metric spaces are Euclidean and Banach spaces. \begin{pmatrix} In the exercises you will see that the case m= 3 proves the triangle inequality for the spherical metric of Example 1.6. Problems for Section 1.1 1. triangle inequality: $h(\bb x+\bb y) \leq h(\bb x) + h(\bb y)$. \] The R code below generates the left and right columns of the figure. Show that the real line is a metric space. /Type /Page Figure B.4.1: \def\c{\,|\,} The concepts of metric and metric space are generalizations of the idea of distance in Euclidean space. 2. As before we write \(x_n\) for the \(n\)th element in the sequence and use the notation \(\{ x_n \}\), or more precisely \[\{ x_n \}_{n=1}^\infty .\] Metric Spaces Then d is a metric on R. Nearly all the concepts we discuss for metric spaces are natural generalizations of the corresponding concepts for R with this absolute-value metric. PROBLEM 4: Prove that if X is Euclidean space @Ann p(r;R) = @B p(r) \@B p(R): PROBLEM 5: Prove that if X is a discrete metric space and A ˆX then @A = ;. This de nes a metric on Rn; which we will prove shortly. This is another example of a metric space that is not a normed vector space: V is a metric space, using the metric de ned from jjjj, and therefore, \[ d(\bb x,\bb y) = \sqrt{\sum_{i=1}^d (x_i-y_i)^2},\] Consider the map f: R !R2 de ned by f(x) = (px 2;px 2). Example 7.4. \def\diag{\operatorname{\sf diag}} Euclidean metric. Similarly, for $\bb x\in B_{\epsilon}(\bb 0)$ the components $x_n$, where $n < N$, satisfy $|x_n| < n\epsilon$. When n= 1;2;3, this function gives precisely the usual notion of distance between points in these spaces. \end{align*}. Then for each $\bb x\in A$ we can construct $B_{\epsilon}(\bb y)$ such that $\bb x\in B_{\epsilon}(\bb y)\subset A$ (taking $\epsilon$ to be sufficiently small). 94 7. \[ \|\bb x\|_{2,\diag(\bb w)} = \sqrt{\sum_{i=1}^d w_i^2 x_i^2}.\]. Pointed metric spaces \vdots & \vdots & \ddots &\vdots\\ As $p$ increases from 1 to $\infty$ the contours change their shape from diamond-shape to square-shape. The metric structure of the Euclidean space simplifies some of the properties described in the previous chapter. The minimum dimension of the Euclidean space in which X is embeddable is denoted by dim X. A sequence in a metric space \((X,d)\) is a function \(x \colon {\mathbb{N}}\to X\). Given a set , we say is a metric space if it comes equipped with a special function that can compute the distance between any two points of . /Parent 8 0 R We prove that each of the above are metric spaces by showing that they are normed linear spaces, where the obvious candidates are used for norms. /Filter /FlateDecode [You Do!] /PTEX.InfoDict 9 0 R \def\bb{\boldsymbol} Then the OPEN BALL of radius >0 Let $G$ be an open set and ${\bb x}\in G$. 1. Euclidean space is considered to be a finite dimensioned space. 1. This is the normal subject of a typical linear algebra course. We’ll give some examples and define continuity on metric spaces, then show how continuity can be stated without reference to metrics. Figure B.4.1 below shows contour lines of four $L_p$ norms in the left column and four $L_{p,W}$ norms in the right column, where $W=\diag(2,1)$. In particular, whenever we talk about the metric spaces Rn without explicitly specifying the metrics, these are the ones we are talking about. An important metric space is the n -dimensional euclidean space R n = R × R × ⋯ × R. We use the following notation for points: x = (x 1, x 2, …, x n) ∈ R n. We also simply write 0 ∈ R n to mean the vector (0, 0, …, 0). 0 & w_2&\cdots&0\\ If $\bb x\in B_{\epsilon}(\bb 0)$ for a given $\epsilon$ then for all $n\in\mathbb{N}$ we have $\min(|x_n|,1) < n\epsilon$. %PDF-1.4 The Euclidean space $\R^d$ is second countable, and in particular one choice for $\mathcal{G}$ in Definition B.1.5 is the set of all open balls with rational centers and radii. \|\bb x\|_{\infty} &= \max\{|x_1|,\ldots,|x_d|\} \qquad \text{ (achieved by letting } p\to\infty). 2. For example, if $\diag(\bb w)$ is an all zero matrix, except for its diagonal This way of defining Euclidean space is still in use under the name of synthetic geometry. Even more interesting are the in nite dimensional cases. Theorem 1.5. the Euclidean space is a metric space $(\R,d)$ (we prove later later in this chapter that the Euclidean distance above is a valid distance function). A metric space (X,d) is said to be complete if every Cauchy sequence in X converges (to a point in X). TASK: Rigorously prove that the space (ℝ2,) is a metric space. These properties are called postulates, or axioms in modern language. \[ \diag(\bb w) = These will be the standard examples of metric spaces. Together with the Euclidean distance 13.1. /PTEX.FileName (./205_script.pdf) Note that above we are denoting the function using bold-face $\bb f$ to indicate that $\bb f(\bb x)$ is a vector. \def\P{\mathsf{P}} METRIC SPACES Math 441, Summer 2009 We begin this class by a motivational introduction to metric spaces. \R\times \cdots \times \R\times (a,b) \times \R\times\cdots. In summary, we have thus demonstrated that in the space $(\R^{\infty},\bar d)$, the set of open sets is equivalent to the set of unions of intersections of a finite number of simple cylinders. Any incomplete space. >> 4 0 obj << A metric space is a pair (X, d), where X is a set and d is a metric on X; that is a function on XX such that for all x, y, z X, we have: (M1)d(x, y)  0. 6 0 obj << Assume that is not sequentially compact. Metric spaces 275 Example 13.12. Discrete metric space is often used as (extremely useful) counterexamples to illustrate certain concepts. # generate right column (weighted Lp norms), symmetry: $g( \bb v,\bb u) = g(\bb u,\bb v)$, bi-linearity: $g(\alpha \bb u+\beta \bb v,\bb w) =\alpha g(\bb u,\bb w) + \beta g( \bb v,\bb w)$, positivity: $h(\bb x)=0$ if and only if $\bb x=\bb 0$, homogeneity: $h(c \,\bb x)=|c|\, h(\bb x)$. /Contents 7 0 R The general definition of a compact space (Definition B.1.6) is hard to verify, but the following simple condition is veru useful for verifying compactness in $\R^d$. /PTEX.PageNumber 125 Knowing whether or not a metric space is complete is very useful, and many common metric spaces are complete. Definition: A function is a metric if it satisfies the following three properties for any choice of elements . Introduction When we consider properties of a “reasonable” function, probably the first thing that comes to mind is that it exhibits continuity: the behavior of the function at a certain point is similar to the behavior of the function in a small neighborhood of the point. The notion of a sequence in a metric space is very similar to a sequence of real numbers. \def\defeq{\stackrel{\tiny\text{def}}{=}} \begin{align} \tag{*} We then have the following fundamental theorem characterizing compact metric spaces: Theorem 2.2 (Compactness of metric spaces) For a metric space X, the following are equivalent: (a) X is compact, i.e. ��-��3&���qf�4� 3`�0p�8������g�s�~��jwB0o��%ܩ�(D����. Chapter 8 Euclidean Space and Metric Spaces 8.1 Structures on Euclidean Space 8.1.1 Vector and Metric Spaces The set K n of n -tuples x = ( x 1;x 2:::;xn) can be made into a vector space by introducing the standard operations of addition and scalar multiplication Non-examples. Theorem. We denote a vector in $\R^d$ when $d>1$ in bold and refer to its scalar components via subscripts. 1. is compact. Show that (X,d) in Example 4 is a metric space. \end{align} And let be the discrete metric. Theorem: R is a complete metric space | i.e., every Cauchy sequence of real numbers converges. Sequences. The concept of compactness (Definition B.1.6) has some important consequences (Propositions B.3.5 and B.3.6). (M3) d(x, y) = d(y, x). For a metric space the following two statements are equivalent. A metric space is separable space if it has a countable dense subset. A closed subset of a complete metric space is a complete sub- space. Theorem 1.6. >> \begin{align*} the corresponding weighted $L_2$ norm is \end{align*} stream b��\��f?ޖ��4g�b3�E�33�����$�)�E:�%�I�ʛ�lYMY6_�{e���u �fo������b��fh�pK�1� For R2 with the Euclidean metric de ned in Example 13.6, the ball B r(x) is an open disc of radius rcentered at x.For the ‘1-metric in Exam- ple 13.5, the ball is a diamond of diameter 2r, and for the ‘1-metric in Exam- ple 13.7, it is a square of side 2r. The following properties of a metric space are equivalent: Proof. The more general $L_p$ norm This space (X;d) is called a discrete metric space. Before making R n a metric space, let us prove an important inequality, the so-called Cauchy-Schwarz inequality. /Length 117 Defn A metric space is a pair (X,d) where X is a set and d : X 2 [0,) ... the standard Euclidean distance as d 2 (x,y ... We will concentrate our studies on the cases p=1,2,. It follows that points in $B_{\epsilon}(\bb 0)$ are an intersection of a finite number of sets of the form 3. xڍ�1 We have the Euclidean ... 2.The usual metric or Euclidean metric on Rnis de ned as in the Motivation section above. Conversely, if $\bb f$ is continuous, for all $\epsilon>0$ there exists $\delta>0$ such that whenever $\|\bb x-\bb y\|^2\leq \delta^2$ we have \[\|\bb f(\bb x)-\bb f(\bb y)\|^2=\sum_{j=1}^k |f_j(\bb x)-f_j(\bb y)|^2 < \epsilon^2.\] This implies $|f_j(\bb x)-f_j(\bb y)|^2 < \epsilon^2$ and $|f_j(\bb x)-f_j(\bb y)| < \epsilon$, implying the continuity of $f_1,\ldots,f_k$. Example 3.3. A non-diagonal matrix $W$ would result in rotated versions of the figures in the right column. For instance, R \mathbb{R} R is complete under the standard absolute value metric, although this is not so easy to prove. (i) The space is separable, (ii) The space is perfectly separable!^2). It is easy to see that the Euclidean metric satis es (1){(3) of a metric. \|\bb x\|_2 &=\|\bb x\| \qquad \text{ (the Euclidean norm)}\\ /Resources 5 0 R /MediaBox [0 0 612 792] Proof: Let fx ngbe a Cauchy sequence. /Length 1775 Example 6: Let V be a normed vector space | for example, R2 with the Euclidean norm. endstream 0 & \cdots & 0& w_d DEFINITION: Let be a space with metric .Let ∈. In the case of $R^{\infty}$, second countability is demonstrated by taking all simple cylinders (see Example B.4.4) whose base $(a,b)$ has rational endpoints and noting that a countable union of sets that are countably infinite is countably infinite. The propositions above confirm that the Euclidean space $(\R^d,d)$, where A proof is available in (Rudin, 1976). 2. is sequentially compact. We de ne V= f( x 1;x Let Cbe the unit circle fx2V jjjxjj= 1g. An important property of complete metric spaces, preserved under homeomorphisms, is the Baire property, on the strength of which each complete metric space without isolated points is uncountable. /BBox [0.00000000 0.00000000 612.00000000 792.00000000] every open covering of X has a finite subcovering. Specifically, must satisfy the axioms of a metric. Example 1.7. This metric structure is commonly referred to in the literature as the product topology of $\R^{\infty}$. PROBLEM 6: Prove that if B is a closed set, then @B ˆB. $. Since $\bar d(\bb x+\bb c,\bb z+\bb c)=\bar d(\bb x,\bb z)$, we also have that $B_{\epsilon}(\bb y)$ is an intersection of a finite number of simple cylinders or sets of the form expressed in (*). endobj Equal height contours of the $L_p$ norm (left column) and weighted $L_{p,W}$ norm with $W=\diag(2,1)$ (right column), in the two dimensional case $d=2$. The convergence problems mentioned in Example~B.4.3 leads to the common practice of defining the metric structure on $\R^{\infty}$ using the distance function $\bar d$ in Example B.4.4 rather than the Euclidean distance. A separable metric space is hereditarily separable^). \|\bb x \|_1 &= \sum_{i=1}^d |x_i|\\ It is easily checked that this is an isometry when R is given the usual metric and R2 is given either the 2-dimensional Euclidean metric, the d 1 metric, or the d 1metric (see Homework 3). The $L_p$ norm can be further generalized as follows. has the following special cases Their great innovation, appearing in Euclid's Elements was to build and prove all geometry by starting from a few very basic properties, which are abstracted from the physical world, and cannot be mathematically proved because of the lack of more basic tools. all metric spaces, saving us the labor of having to prove them over and over again each time we introduce a new class of spaces. >> endobj By Proposition B.1.2 there exists $r>0$ for which $B_r({\bb x})\subset G$. For a finite metric space X with elements p 0, p 1, …, p n, let D i, j = d (p i, p j) 2 and g i, j = 1 2 (D 0, i + D 0, j − D i, j). 7 0 obj << The Euclidean norm is the most popular norm. w_1 &0 &\cdots& 0\\ Metric Spaces §1. Here we can think of the f(R) as a copy of R living inside of R2. �0E��"h����]G���T��_E:�$���?��JȃE��!Җ�0p �QC���_���5�n�J��VG��C�Đ@q�Lm��5?����H�֚�v}?�=_8q4� This metric is called the Euclidean metric and (Rn;d) is called Euclidean space. Examples. While the Hilbert space is an extension of Euclidean space and an infinite dimensioned space. A subset A of a metric space X is called open in X if every point of A has an -neighbourhood which lies completely in A. On the other hand, let $A$ be an intersection of a finite number of simple cylinders. Then V (a) (0, 1). is a metric on Rn, called the Euclidean metric. \|\bb x\|_p=\left(\sum_{i=1}^d |x_i|^p\right)^{1/p}, \qquad p\geq 1, Note that in the first and third cases above, we have $k=1$, and in the last case above, we have $k=d=1$. Metric spaces and metrizability 1 Motivation ... we have talked about Rn usual and how relatively easy it is to prove things about it due to the fact that the topology is de ned by a distance function. Theorem 4. Open (Closed) Balls in any Metric Space (,) EXAMPLE: Let =ℝ2 for example, the white/chalkboard. is a metric space. Define d: R2 ×R2 → R by d(x,y) = (x1 −y1)2 +(x2 −y2)2 x = (x1,x2), y = (y1,y2).Then d is a metric on R2, called the Euclidean, or ℓ2, metric.It corresponds to Since for every real number there is a rational number that is arbitrarily close, we can select $B_{r'}({\bb x}')$ where $r',{\bb x}'$ are rationals such that ${\bb x}\in B_{r'}({\bb x}') \subset B_r({\bb x})\subset G$. /Matrix [1.00000000 0.00000000 0.00000000 1.00000000 0.00000000 0.00000000] 3. is complete and totally bounded. Turns out, these three definitions are essentially equivalent. Let Xbe any non-empty set and let dbe de ned by d(x;y) = (0 if x= y 1 if x6= y: This distance is called a discrete metric and (X;d) is called a discrete metric space. The metric structure of the Euclidean space simplifies some of the properties described in the previous chapter. Proof Choose < min {a, 1-a}. Now we’ll prove that R is a complete metric space, and then use that fact to prove that the Euclidean space Rn is complete. ���T�x:��" �.�[�?\���ga&�����z�(%��* The Euclidean space could accommodate almost all functions but was limited in dimensions i.e. We want to endow this set with a metric; i.e a way to measure distances between elements of X.A distanceor metric is a function d: X×X →R such that if we take two elements x,y∈Xthe number d(x,y) gives us the distance between them. Metrics and Metric Spaces.