Balls in metric and normed spaces

Show that the open unit ball in $\mathbb {R}^{n}$ is an open subset of $\mathbb {R}^{n}$.

Prove that every closed ball in a metric space is closed.

Prove that if $V$ is a normed vector space, $f \in V$, and $r > 0$, then

Let $\left\| \cdot \right\|_{1},\left\| \cdot \right\|_{2}$ be norms on the linear space $X$, and $\bar{B}_{i}(0,1)$ be the unit closed ball in $\left(X,\left\| \cdot \right\|_{i}\right), i = 1,2$. Prove that if $\bar{B}_{1}(0,1) = \bar{B}_{2}(0,1)$, then $\left\| x\right\|_{1} = \left\| x\right\|_{2}, x \in X$.

Let a function $\varphi$ on a linear space $X$ satisfy the first two axioms of the norm. Prove that $\varphi$ is a norm on $X$ if and only if the set $B = \left\{ x \in X \mid \varphi (x) \leq 1\right\}$ is convex.

Is it true that in an arbitrary metric space the closure of the open ball $B(x, r)$ is the closed ball $\bar{B}(x, r)$?

Give an example of a metric space and balls $B\left(x, r_{1}\right)$ and $B\left(y, r_{2}\right)$ in it such that $r_{1} > r_{2}$, but $B\left(x, r_{1}\right) \varsubsetneqq B\left(y, r_{2}\right)$.

Prove that in the space $\mathbb {Q}_{p}$ the triangle axiom holds in a strengthened form: $\rho (x, y) \leqslant \max \left\{ \rho (x, z), \rho (y, z)\right\}$, and moreover, among the three numbers $\rho (x, y), \rho (x, z)$ and $\rho (y, z)$ two are necessarily equal (all "triangles" in this space are isosceles).

Prove that in the space $\mathbb {Q}_{p}$:

Prove that:

In the ball $B(0,1)$ of the space $l_{2}$, place a countable number of disjoint balls of radius $\frac{1}{10}$.

Prove that for any two distinct points of a metric space there exist disjoint balls centered at these points.

Can a ball of radius 4 be a proper subset of a ball of radius 3?

Describe open and closed balls in the metric space $\left(\mathbb {R}^{2}, d\right)$, where the distance $d$ is defined by $d\left(\left(x_{1}, x_{2}\right),\left(y_{1}, y_{2}\right)\right):=\left(\left|x_{1}-y_{1}\right|^{3}+\left|x_{2}-y_{2}\right|^{3}\right)^{1 / 3}$.

Describe open and closed balls in the metric space ( $C[0,1], d$ ), where $d(f, g):=\sup \left\{ \left|f(x)-g(x)\right|: x \in [0,1]\right\}$.

Let $u(x)=\sin x$ and $v(x)=0$ and let $C[0,2 \pi ]$ denote the metric space of continuous functions on $[0,2 \pi ]$ with the distance $d(f, g):=\sup \left\{ \left|f(x)-g(x)\right|: x \in [0,2 \pi ]\right\}$. Give an example of a function that belongs to both $B(u, 2 / 3)$ and $B(v, 2 / 3)$.

In the metric space $(\mathbb {R}, d)$ where

describe open balls $B(0, r)$ for:

Prove that an open ball in a metric space is an open set.

Prove that a closed ball in a metric space is a closed set.

Prove that in a metric space ( $M, d$ ) if $a \in M$ and $r>0$ then the set $\left\{ x: d(x, a)=r\right\}$ is closed.

Let $X$ be a normed space. Prove that the closed unit ball is a closed subset of $X$.

Let $X$ be a normed space. Prove that the closed unit ball is the closure of the open unit ball.

Let $X$ be a normed space. Prove that the unit ball of $X$ has a unique center and radius.

Show that in a metric space an open ball is open.

In a metric space $X$, is it possible for $r > 0$ and two distinct points $u$ and $v$ in $X$ to have $B(u, r) = B(v, r)$ ? Is this possible in Euclidean space $\mathbb {R}^{n}$ ? Is it possible in a normed linear space?

Let $X$ be a metric space, $x$ belong to $X$ and $r > 0$.

Let $(V,\left\| \cdot \right\| )$ be a normed linear space. Let $S = \left\{ v:\left\| v\right\| < 1\right\}$. Show that $S$ is an open set in $V$.