Tables Collection

Topological Properties of common Metric Spaces

Topological Properties of common Metric Spaces
Metric Space		Norm	Distance	Separability	Completeness (Banach space for normed space)
$n$-dimension Eucilidean space $(\mathbb{R}^n,d)$		$$\Vert x\Vert = \sqrt{\sum_{i=1}^{n} x_i^2}$$	$$ d(x,y) = \sqrt{\sum_{i=1}^{n} (x_i-y_i)^2} $$	✓	✓
Discrete metric space $(X,d_0)$	$X$ is countable	/	$$d_0(x,y)=\begin{cases}0, x=y\\1, x\not=y\end{cases}$$	✓	✓
Discrete metric space $(X,d_0)$	$X$ is uncountable	/	$$d_0(x,y)=\begin{cases}0, x=y\\1, x\not=y\end{cases}$$	✗	✓
Space of continuous functions $\mathscr{C}[a,b]$		$$\Vert f\Vert = \max\limits_{x\in [a,b]} \vert f(x)\vert$$	$$d(f,g) = \max\limits_{x\in [a,b]} \vert f(x)-g(x)\vert$$	✓	✓
Space of continuous functions $\mathscr{C}[a,b]$		$$\Vert f\Vert = \int_a^b \vert f(t)\vert \,\mathrm{d}x$$	$$d(f,g) = \int_a^b \vert f(t)-g(t)\vert \,\mathrm{d}x$$	✓	✗
Space of bounded sequences $\ell^\infty$		$$\Vert x\Vert = \sup\limits_{i\geq 1} \vert x_i\vert$$	$$d(x,y) = \sup\limits_{i\geq 1} \vert x_i-y_i\vert$$	✗	✓
Variable exponent sequence space $\ell^p$		$$\Vert x\Vert = \left(\sum_{i=1}^{\infty} \vert x_i\vert^p\right)^\frac{1}{p}$$	$$d(x,y) = \left(\sum_{i=1}^{\infty} \vert x_i-y_i\vert^p\right)^\frac{1}{p}$$	✓	✓
$p$-norm Lebesgue space $L^p$		$$\Vert f\Vert = \left(\int_{[a,b]} \vert f(x)\vert^p \,\mathrm{d}x\right)^\frac{1}{p}$$	$$d(f,g) = \left(\int_{[a,b]} \vert f(x)-g(x)\vert^p \,\mathrm{d}x\right)^\frac{1}{p}$$	✓	✓