# Questions tagged [metric-spaces]

Metric spaces are sets on which a metric is defined. A metric is a generalization of the concept of "distance" in the Euclidean sense. Metric spaces arise as a special case of the more general notion of a topological space.

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### Show that the set of differentiable functions is a complete metric space, where $d(f,g)=\max_{x\in[0,1]}\{|f(x)-g(x)|+|f'(x)-g'(x)|\}$

$f$ and $g$ are in $C^1([0,1])$. I can show that the space of functions, along with this metric, is indeed a metric space. But showing that it's complete is proving to be a bit more complicated. Say I'...
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### Using calculus to show that $f_n(x)=x^n$ is not Cauchy in $C^0[0,1]$

As the title says, I need to prove "using calculus" that the sequence of functions $f_n(x)=x^n$ is not Cauchy in $C^0[0,1]$. The thing that came to my mind is to use the $L_1$ or $L_2$ norm ...
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### What can be concluded about $f\$?

Let $X$ and $Y$ be metric spaces. Assume that $Y$ is a discrete metric space and that $f : X \longrightarrow Y$ is a contraction. What can be concluded about $f\$? What I observe is that if there ...
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### Converse of contraction mapping theorem

Let $(X,d)$ be a metric space such that, if $Y \subset X$ is closed, then every contraction mapping on $Y$ has a fixed point. Show that $X$ is complete. This problem appeared in an exam and I ...
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### $(1+x)^d \leq 1 + x^d$ for $x \geq 0$ and $d \in (0,1]$

I come across the following inequality: for all $x \geq 0$ and $d \geq 0$, if $d \leq 1$: $$(1+x)^d \leq 1 + x^d,$$ if $d \geq 1$: $$(1+x)^d \geq 1 + x^d.$$ I think they are related to convex ...
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### Example of a set that is neither open nor closed in given metric space $M$

I gave this example: $a\in M$, take the closed ball $B[a,r]=\{x\in M | d(a,x)\leq r\}$, take an element $x$ such that $d(a,x)=r$ and do $S=B[a,r]-\{x\}$. It's easy to prove that this set is neither ...
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### Open balls with center $\pm\infty$ in the extended real line
Define a function $f$ from $\overline{\mathbb{R}}$ to the interval $[-1,1]$ as follows: For $x\in\mathbb{R}$, let $f(x)=x/(1+|x|)$; on the other hand, write $f(\infty)=1$ and $f(-\infty)=-1$. Then \$d(...