From 28407333ffceca9b99fae721c30e8ae146a863da Mon Sep 17 00:00:00 2001 From: Justin Gassner Date: Wed, 14 Feb 2024 07:24:38 +0100 Subject: Update --- pages/general-topology/metric-spaces/index.md | 215 ++++++++++++++++++++++++++ 1 file changed, 215 insertions(+) create mode 100644 pages/general-topology/metric-spaces/index.md (limited to 'pages/general-topology/metric-spaces/index.md') diff --git a/pages/general-topology/metric-spaces/index.md b/pages/general-topology/metric-spaces/index.md new file mode 100644 index 0000000..c0dc45a --- /dev/null +++ b/pages/general-topology/metric-spaces/index.md @@ -0,0 +1,215 @@ +--- +title: Metric Spaces +parent: General Topology +nav_order: 8 +has_children: true +has_toc: false +--- + +# {{ page.title }} + +{% definition Metric, Metric Space %} +A *metric* (or *distance function*) on a set $X$ is +a mapping $d : X \times X \to \RR$ with the properties \ +**(M1)** $\ \forall x,y \in X : d(x,y) = 0 \iff x=y \quad$ *(point separation)* \ +**(M2)** $\ \forall x,y \in X : d(x,y) = d(y,x) \quad$ *(symmetry)* \ +**(M3)** $\ \forall x,y,z \in X : d(x,z) \le d(x,y) + d(y,x) \quad$ *(triangle inequality)* \ +We say that $d(x,y)$ is the *distance* between $x$ and $y$. \ +A *metric space* is a pair $(X,d)$ consisting of a set $X$ +and a metric $d$ on $X$. +{% enddefinition %} + +Setting $x=z$ in **(M3)** and applying **(M1)** & **(M2)** +gives us $0 = d(x,x) \le 2 d(x,y)$, hence $d(x,y) \ge 0$. +This *nonnegativity* of the metric is often part of the definition. + +Relaxing **(M1)** to the condition $\forall x \in X : d(x,x) = 0$ +leads to the notion of a *semi-metric* +and that of a *semi-metric space*. +Nonnegativity still follows as shown above. + +*Pseudo-metric* is usually a synonym for *semi-metric*. + +*Quasi-metric* refers to dropping **(M2)**. + +An *ultrametric* satisfies in place of **(M3)** the stronger condition +$d(x,z) \le \max \braces{d(x,y),d(y,z)}$. + +{% definition Metric Subspace %} +A *metric subspace* of a metric space $(X,d)$ is a pair $(S,d_S)$ +where $S$ is a subset of $X$ and +$d_S$ is the restriction of $d$ to $S \times S$. +{% enddefinition %} + +Clearly, a metric subspace of a metric space is itself a metric space. + +{% proposition %} +Let $(X,d)$ be a (semi-)metric space. +- For all $x,y,z \in X$ we have the *inverse triangle inequality* + + $$ + \abs{d(x,y) - d(y,z)} \le d(x,z). + $$ + +- For all $v,w,x,y \in X$ we have the *quadrilateral inequality* + + $$ + \abs{d(v,w) - d(x,y)} \le d(v,x) + d(w,y) + $$ +{% endproposition %} + +The proofs are straightforward. + +TODO +- isometry +- metric induced by a norm +- metric product + +{% definition Diameter %} +The *diameter* of a subset $S$ of a metric space $(X,d)$ is the number + +$$ +\diam{S} = \sup \braces{d(x,y) : x,y \in S} \in \braces{-\infty} \cup [0,\infty]. +$$ +{% enddefinition %} + +Note that $\diam{S} = -\infty$ iff $S = \varnothing$, +and $\diam{S} = 0$ iff $S$ is a singleton set. + +{% definition Distance %} +Suppose $(X,d)$ is a metric space. +The *distance* from a point $x \in X$ to a subset $S \subset X$ is + +$$ +\dist{x,S} = \inf \braces{d(x,y) : y \in S} \in [0,\infty]. +$$ +{% enddefinition %} + +Note that $\dist{x,S} = \infty$ iff $S = \varnothing$. + +{% definition Convergence, Limit, Divergence %} +Let $(X,d)$ be a metric space. +A sequence $(x_n)_{n \in \NN}$ in $X$ is said to *converge to a point $x \in X$*, if + +$$ +\forall \epsilon > 0 \ \ \exists N \in \NN \ \ \forall n \ge N : d(x,x_n) < \epsilon. +$$ + +In this case, we call $x$ a *limit (point)* of the sequence. +Symbolically this is expressed by + +$$ +\lim_{n \to \infty} x_n = x +$$ + +or by saying that $x_n \to x$ as $n \to \infty$. + +We call a sequence in $X$ *convergent* +if it converges to some point of $X$ +and *divergent* otherwise. +{% enddefinition %} + +For a semi-metric space the definition remains the same. +However, the notation $\lim x_n = x$ can be misleading, +because there might be more than one limit point. + +{% proposition %} +A sequence in a metric space has at most one limit. +{% endproposition %} + +In other words: The limit of a convergent sequence in a metric space is unique. + +{% proof %} +Let $(x_n)$ be a convergent sequence in a metric space $(X,d)$ with limit point $x$. +If $x'$ is another limit point of $(x_n)$, +then $d(x,x') \le d(x,x_n) + d(x_n,x')$ for all $n \in \NN$ by **(M3)**. +Given $\epsilon >0$, there exist natural numbers $N$ and $N'$ such that +$d(x,x_n) < \epsilon$ for all $n \ge N$ and +$d(x,x_n) < \epsilon$ for all $n \ge N'$. +Both hold, if $n$ is large enough ($\ge \max \braces{N,N'}$ to be precise). +It follows that $d(x,x') < 2 \epsilon$. +Since $\epsilon$ was arbitrary, $d(x,x') = 0$. +Therefore, $x=x'$ by **(M1)**. +{% endproof %} + +{% corollary %} +A semi-metric space $X$ is a metric space if and only if +every sequence in $X$ has at most one limit. +{% endcorollary %} + +{% definition %} +Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. +A mapping $f: X \to Y$ is called +- *continuous at a point $x \in X$* if + + $$ + \forall\epsilon>0 \ \ \exists\delta>0 \ \ \forall x' \in X : \big\lparen d_X(x,x') \le \delta \implies d_Y(f(x),f(x')) < \epsilon \big\rparen + $$ + +- *continuous* if it is continuous at every point of $X$, that is + + $$ + \forall x \in X \ \ \forall\epsilon>0 \ \ \exists\delta>0 \ \ \forall x' \in X : \big\lparen d_X(x,x') \le \delta \implies d_Y(f(x),f(x')) < \epsilon \big\rparen + $$ + +- *uniformly continuous* if + + $$ + \forall\epsilon>0 \ \ \exists\delta>0 \ \ \forall x,x' \in X : \big\lparen d_X(x,x') \le \delta \implies d_Y(f(x),f(x')) < \epsilon \big\rparen + $$ + +- *Lipschitz continuous* if + + $$ + \exists L \ge 0 \ \ \forall x,x' \in X : d_Y(f(x),f(x')) \le L \, d_X(x,x') + $$ +{% enddefinition %} + +{% definition Open Ball, Closed Ball, Sphere %} +Suppose $(X,d)$ is a metric space. +The *open ball* with center $c \in X$ and radius $r>0$ is the set + +$$ +B(c,r) = \braces{x \in X : d(x,c) < r}. +$$ + +The *closed ball* with center $c \in X$ and radius $r>0$ is the set + +$$ +\overline{B}(c,r) = \braces{x \in X : d(x,c) \le r}. +$$ + +The *sphere* with center $c \in X$ and radius $r>0$ is the set + +$$ +S(c,r) = \braces{x \in X : d(x,c) = r}. +$$ +{% enddefinition %} + +Observe that $S(c,r) = \overline{B}(c,r) \setminus B(c,r)$. + +{% definition Open Subset of a Metric Space %} +A subset $O$ of a metric space is called *open* if for every point $x \in O$ +there exists an $\epsilon > 0$ such that $B(x,\epsilon) \subset O$. +{% enddefinition %} + +{% proposition Metric Topology %} +Let $(X,d)$ be a metric space. +The collection of open subsets of $X$ forms a topology on $X$. +This topology is called the *metric topology* on $X$ induced by $d$. +{% endproposition %} + +{% proposition %} +- Open balls in a metric space are open with respect to the metric topology. +- Closed balls in a metric space are closed with respect to the metric topology. +- The boundary (with respect to the metric topology) of an open or closed ball + is the sphere with the same center and radius. Not true!!!! +- The collection of open balls in a metric space forms a basis of the metric topology. +{% endproposition %} + +## Complete Metric Spaces + +- Definition +- Banach Fixed-Point Theorem +- Baire +- Metric Completion -- cgit v1.2.3-54-g00ecf