Metric spaces are sets on which a metric is defined. A metric is a generalisation of the concept of "distance". Metric spaces should not be confused with topological spaces.

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Erwine Kryszeg's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS, Section 1.5-8

In Section 1.5-8, in his book, INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS, Kryszeg tries to show that the set $X$ of all polynomials defined on a given closed interval $[a,b]$ on the real ...
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Finding a norm on $ \mathbb{R}^X $ such that the “natural” embedding of a metric space $ X $ in $ \mathbb{R}^X $ becomes an isometry

This question is related to this one: Every metric space can be isometrically embedded in a Banach space, so that it's a linearly independent set Kuratowski's embedding indeed seems to be the ...
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Open sets in topology and metric spaces

Let $\tau$ = { $\emptyset,[0,1],\mathbb{R}$} ($\mathbb{R},\tau$) is a topological space, right? Since the intersection of any of the sets in $\tau$ is itself in $\tau$, and same for the union. But ...
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Integral equation and metric spaces

Let $C([0,\frac{\pi }{2}])$ be the set off all continuous functions defined on $[0,\frac{\pi }{2}]$ . Prove that this integral equation $$ f(t) = \int\limits_0^{\frac{\pi }{2}} {\arctan } ...
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Existence of a mapping in a nonseparable Banach space that moves all nearby points to far-away points

Does there exist a nonseparable Banach space $X$, a mapping $F: X\to X$, and an open nonempty subset $D\subset X$ such that $$ \forall\,E>0 \quad \exists\,\delta>0: \quad \forall\,x,y\in D \quad ...
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k-Cells are Connected

I am studying real analysis from Baby Rudin, and while the book proves that real intervals are connected, it does not say anything regarding k-cells. I would expect them to also be connected, but do ...
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260 views

Metrics on X. Show that they are equivalent if and only if…

Suppose that $d$ and $ρ$ are metrics on a set $X$. Prove the following statement: The metrics $d$ and $ρ$ are equivalent if and only if the class of $d$-open sets of $X$ exactly coincides with the ...
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Show that the metric $d_1$ on $C([0,1])$ does not give rise to a complete metric space. [duplicate]

Show that the metric $d_1$ on $C([0,1])$ does not give rise to a complete metric space. $$ d_1(f,g) = \int_0^1 |f(s)−g(s)| \, ds $$
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metric characterization for connectedness

Is there a metric characterization of connectedness? I'm looking for something like the following metric characterization of compactness: A metrizable topological space is compact if, and only if, ...
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Endowing an abelian group with a metric.

I solved the following exercise, which is not hard: Let $G$ be an additive abelian group, such that exists $f: G \to \mathbb{R}$ satisfying: $f(0) = 0$ and $f(x) > 0$ for all $x \neq ...
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Connected spaces of $M(n,\mathbb R)$

Consider $M(n,\mathbb R)$, the space of all $ n\times n$ matrices over $\mathbb R$.Which of the following are connected? a.$O(n)$ the set of all orthogonal matrices b.$GL(n,\mathbb R)$ set of all ...
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How is $\sqrt{2}$, for example, in the closure of $\mathbb{Q}$ in the usual metric space $\mathbb{R}$?

Let $\mathbb{R}$ be the set of all real numbers under the usual metric $d$ defined as follows: $$d(x,y) \colon= |x-y|$$ for all $x$, $y$ in $\mathbb{R}$, and let $\mathbb{Q}$ be the set of all ...
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92 views

Compact subsets of $M(n,\mathbb R)$

Consider $M(n,\mathbb R)$, the space of all $ n\times n$ matrices over $\mathbb R$.Which of the following are compact? a.$O(n)$ the set of all orthogonal matrices b.$GL(n,\mathbb R)$ set of all ...
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1answer
42 views

Are there properties of vector space equipped with two norms?

I am interested in a vector space equipped with two norms$ \lvert \lvert \cdot \rvert \rvert$ and $ \lvert \lvert \cdot \rvert \rvert ^*$ satisfies that there is $M>0$ such that $ \lvert \lvert x ...
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Equivalance of norms

Let $X$ be the vector space of all real valued functions defined on $[0,1]$ having continuous first-order derivatives. How to show that the following norms are equivalent: $\|f\|_1 = |f(0)| + ...
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How to decide completeness of $\ell^\infty$?

Let $\ell^\infty$ denote the set of all bounded sequences $x \colon = (\xi_j)_{j=1}^\infty$, $y \colon= (\eta_j)_{j=1}^\infty$ of complex numbers with the metric $d$ defined as follows: $$ d(x,y) ...
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223 views

Calculate distance in x,y from center based on distance and degrees.

I'm terribly sorry if this question is written like a 5-year old.. But that's the level I'm on in terms of math and coordinate calculations. (Just realized I don't even know what to tag this question ...
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path metrics without geodesics

This is a follow-up of this question. Recall that a metric space $(X,d)$ is called a path-metric space if the distance between any two points in $X$ equals the infimum of lengths of paths between ...
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Is the following function uniformly continuous?

I am supposed to prove if the function $ f:X \to \mathbb {R}, f (x) = dist (x, A)$ where $ A$ is an arbitrary subset o the metric space $X $ is uniformly continuous. If both points $ x $ and $ y $ ...
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Every metric space can be isometrically embedded in a Banach space, so that it's a linearly independent set

I'm trying to prove the following: Every metric space can be isometrically embedded in a Banach space, so that it's a linearly independent set. I came up with the following idea: Let $ (X,d) $ be a ...
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Prove Contraction Mapping

The following is given: Eucliden metric $d$, defining the distance between vector $v_1=(x_1,y_1)$ and $v_2=(x_2,y_2)$: $d(v_1,v_2)=\sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}$ $M $ is a mapping of $\mathbb ...
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$C([a,b] \times [c,d],X)$ compared to $C([a,b],C([c,d],X))$ and $C([c,d],C([a,b],X))$

Let $C(Y,X)$ be the space of continuous functions from $Y$ to $X$ together with the supremums norm. Here $Y$ is a compact space and $X$ a metric space. Let $a,b,c,d \in \mathbb R$ be finite, with ...
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Metric Spaces, Continuity and Preference Relations

Let X be a metric space and $\succeq$ be a preference relation on X. The preference relation is continuous if the sets $\succeq (y) =\{x: x \succeq y\}$ and $\preceq (y) = \{x : x \preceq y\}$ are ...
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How to prove continuity of addition over weird metric? Edit: Ignore this. Errors in the problem definition.

Let $f: R \times R \rightarrow R$ and let the metric over $R$ be $d(x,y)=|x-y|$ and let the metric in $R \times R$ be $d_2((x,y),(a,b))= ((x-y)^2+(a-b)^2)^{1/2}$. I believe I understand how to ...
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Finding open balls in $\mathbb{R}^2$

Consider the set $U = \{(x, y) \in \mathbb R^2: y > 0\}$. Working in the metric space $(\mathbb R^2, d_E)$, find open balls $B_1, B_2, B_3,\ldots$ with $U = \bigcup_{i \in \mathbb{N}} B_i$. Then ...
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180 views

shortest path in complete metric space

Let $(X,d)$ be a complete connected by arcs metric space. We define the length of a continuous path $\gamma: [0,1] \rightarrow X$ to be \begin{equation*} \sup\limits_{0=a_{0}<a_{1}<... a_{n}=1} ...
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When are the following inclusions $\subsetneq$

When does the "equality" part of inclusion fail in: $$\overline{A \cap B} \subseteq \overline{A} \cap \overline{B}$$ and $$Int(A \cup B) \supseteq Int(A) \cup Int(B)$$ ? Can you provide an simple ...
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continuity of a metric d

from Continuity of the Metric and Convergence Sequences, why $d^{-1}(V)$ is an open ball? to be an open ball, I think it contains elements of $X$, not $X^{2}$. why is it?
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Showing that $f$ continuous

Let $A$ be a compact subset of a metric space $(X, d)$. Consider the function $f : X → R$ given by $f(x) := $ sup $ \{d(x, y) : y ∈ A\}$ . Show that $f$ is continuous. I tried taking an open subspace ...
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$ {\|f\|}_p = \sqrt[p]{\int_{a}^{b} |f(x)|^p {\rm d}x}$ is a norm

Consider the space $C([a,b])$ of all continuous functions $f\colon [a,b]\rightarrow \mathbb{R}.$ Show that the function $\|\cdot\|_p\colon C([a,b]) \rightarrow [0,\infty),p>1$, given by $$ ...
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Prove $\left|\sum_{i=1}^n x_i y_i \right| \le \dfrac{1}{a} \sum_{i=1}^n {x_i}^2 + \dfrac{a}{4}\sum_{i=1}^n {y_i}^2$

If $X,Y$ are vectors in $\mathbb{R}^n$ and $a>0$ show that: $$\left|\sum_{i=1}^n x_i y_i \right| \le \dfrac{1}{a} \sum_{i=1}^n {x_i}^2 + \dfrac{a}{4}\sum_{i=1}^n {y_i}^2 (*)$$ I started with ...
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Minimal conditions for $\widetilde{d}$ to be metric

Let $(X,d)$ be an arbitrary metric space and $f:[0,\infty) \rightarrow [0,\infty)$ What are the minimal conditions for function $f$ in order $\widetilde{d} = f \circ d: X \times X \rightarrow ...
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Is $\phi$ a norm of E?

Let $(E, \| \|)$ be a normed space. We define $\phi:E \rightarrow [0,\infty)$ as follows: $$\phi(e)= \dfrac{\|e\|}{1+\|e\|}$$ Is $\phi$ a norm of $E$? Please help! Thank you! P.S. This question ...
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Non-Banach, completely metrizable normed vector space

Does there exist a normed vector space $(X,\|\cdot\|)$ over $\mathbb R$ or $\mathbb C$ such that the metric induced by the norm $(x,y)\mapsto\|x-y\|$ is not complete; but there exists some other ...
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If $\inf \{ d(x,y)\mid y \in C \}=0$, then $d(x,z_n)< \frac{1}{n}$

I'm studying a proof I learned in class and I don't quite understand this statement. Let $X$ be a metric space and $C \subset Z$ a nonempty closed set. For each $x \in X$ define $f_{c}(x)=$ inf ...
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Lebesgue number lemma fails for the plane

I need to show that Lebesgue number lemma fails for the plane. I am clueless how to show this. Lemma : Let $(X, d)$ be a compact metric space. Then given an open cover $\mathcal{A}$ of $X$, ...
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Does every compact metric space contain a sequence such that every point of the space is a limit point? [closed]

Let $(X,d)$ a compact metric space. Now, I would like to know if there exists a sequence such that each point in $X$ is a Limit point of this sequence?
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Formal proof that diameter of subset is bounded by diameter of superset (in metric space)

I am asked to show that if $A \subset B$, then $\delta(A)\leq \delta(B)$, where $\delta(A)=\sup_{x,y \in A}d(x,y)$ is the diameter for the non-empty set A in the metric space $(X,d)$. The fact that ...
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Proving a complete and totally bounded metric space is compact.

I'm having trouble writing down the details of this proof formally. Statement: Suppose $(X, d)$ is a metric space that is complete, and totally bounded (i.e., for every $\epsilon > 0$, ...
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Relation between connected subset and clopen subset of a metric space?

I've read that for $A$ a connected subset of a metric space $M$ and $C$ a clopen (closed and open) subset of $M$, one could prove that either $A \subset C$ or $A \cap C=\varnothing$ and use it to ...
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Sequence, subsequences, sub-subsequences in metric spaces

Let $(X,d)$ be a metric space, and let $\{x_n\}_{n \in \mathbb{N}}$ be a sequence in $X$. Assume that every subsequence of $\{x_n\}_{n \in \mathbb{N}}$ has a sub-subsequence that converges to the same ...
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Calculate X Y Z from two specific degrees on a sphere

I am a programmer, don't know much about advanced math. I would need the exact formula(s) that could achieve this, so I can translate it to my programming language. I am having a headache trying to ...
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A property of compact subsets of metric spaces

Let $(X,\varrho)$ be a metric space and $K\subset X$ compact. Then, for every $\,\varepsilon > 0$, $\,K$ can be covered with a finite number of balls of radius $\varepsilon$. Show that the ...
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Defining metrics as a function to something other than the reals.

Generally speaking, a metric for a space R is defined as a function from RxR -> Reals, but does it have to be? Can we define it in more generic terms such as a function from R to a field with certain ...
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Is $A$ compact, $f(A)$ uniformly continuous and is $f^{-1}$ continuous?

$X$ and $Y$ are metric spaces, $A\subseteq X$, $A$ is bounded. map $f:X\to Y$ is continuous. Questions: Is $A$ necessarily compact? Is $f(A)$ uniformly continuous? If given that $f$ is a ...
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How do you prove that Z (the set of integers) equipped with the Euclidean metric (induced from real numbers) is complete?

How do you prove that $\mathbb{Z}$ (the set of integers) equipped with the Euclidean metric (induced from real numbers) is complete? I am having trouble with this question, I don't really know where ...
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On Equivalent Norms in an Infinite Dimensional Vector Space

How many non-equivalent norms can we define in an infinite dimensional vector space? Is there any explicit expression?
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How to define distance between two functions in a non-linear space (example of non-linear space: shape space)?

Suppose I have two parametric circle $f_1=(acost,asint)$ and $f_2=(bcos t,bsint)$, $t\in(0,2\pi),a>0,b>0$, which lies in some non-linear space. Are there any way, how to define the ...
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84 views

When a metric space is a normed space?

I'm trying to figure out that which condition should be provided for a metric space to be normed also?
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If $X$ is locally compact, second countable and Hausdorff, then $X^*$ is metrizable and hence $X$ is metrizable

I need to show that: If $X$ is locally compact, second countable and Hausdorff, then $X^*$ is metrizable and hence $X$ is metrizable. I have already showed that every locally compact Hausdorff space ...