Here’s problem 4 immediately following section 2.3 in Erwine Kryszeg’s book, Introductory Functional Analysis With Applications: Show that in a normed space $X$, vector addition and scalar multiplication are continuous operations with respect to the norm; that is, the mappings defined by $(x,y) \mapsto x+y$ and $(\alpha,x) \mapsto \alpha x$ are continuous. Now the map […]

It is not hard to find proofs showing that $\mathbb{Q}$ is not complete with respect to the metric induced by the valuation $|\;\;|_p$. For example, it is enough to recall that every complete metric space is Baire and to show that $(\mathbb{Q},|\;\;|_p)$ is not Baire. Another option is to show that $\mathbb{Q}_p$ is uncountable, so […]

Let $(M, d)$ be a metric space. I define a translation on $M$ to be a function $f$ from $M$ to $M$ such that $d(x,f(x))=d(y,f(y))$ for all $x$ and $y$ in $M$. In a previous question, I asked if every translation was an isometry in the post Are all metric translations isometries. The answer was […]

I’m trying to prove the following: Show that a subset of a topological space is closed if and only if it contains all of its limit points. Is my proof valid? Definition of limit point: $p$ is a limit point of a subset, if every neighborhood of $p$ contains a point in the subset other […]

In Irving Kaplansky’s “Set Theory and Metric Spaces”, exercise 17 on page 71 asks for an example of a metric space which is isometric to a proper subset of itself. Any infinite discrete space and any $\ell^p$ space are such spaces, for different reasons. I want some kind of middle ground, that is, some example […]

Can a neighbourhood of a point be a singleton set containing that point only ? I think yes.

In $\mathbb{R}^2$ sketch B((1,2),3), the open ball of radius 3 at the point (1,2) with the following metric…. $d(x,y)= \dfrac{5||x-y||_2}{1 + ||x-y||_2} $ I know what the sketch looks like but I don’t know how to compute it. Please help

I’m studying analysis, but I want to know more concepts about metric spaces, so I try to read some explanations in topology books. My textbook is Munkres’ Topology (second edition). I cite some definitions used in this book for convenience. Compact: Every open covering $\mathcal{A}$ of $X$ contains a finite subcollection that also covers $X$. […]

Let $X$ and $Y$ be two open sets in $\mathbb{R}^2$, with $X\subsetneq Y$. Is it possible that $\text{Area}(Y\setminus X)=0$? Is it possible that $\text{Area}(Y\setminus Closure[X])=0$?

Definitions: A geodesic space is a metric space $(X,d)$ such that every two points $x,y \in X$ can be joined by a geodesic. (A path $[0,1] \to X$ is a rectifiable curve if and only if it has a parametrization for which it is Lipschitz continuous. A rectifiable curve is a geodesic if and only […]

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