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 / $$d_0(x,y)=\begin{cases}0, x=y\\1, x\not=y\end{cases}$$ ✓ ✓ \(X\) is uncountable ✗ ✓ Space of continuous functions \(\mathscr{C}[a,b]\) $$\Vert f\Vert = \max\limits_{x\in [a,b]} \vert f(x)\vert$$ $$d(f,g) = \max\limits_{x\in [a,b]} \vert f(x)-g(x)\vert$$ ✓ ✓ $$\Vert f\Vert = \int_a^b \vert f(t)\vert \,\mathrm{d}x$$ $$d(f,g) = \int_a^b \vert f(t)-g(t)\vert \,\mathrm{d}x$$ ✓ ✗ 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 \(L^p\) $$\Vert f\Vert = \left(\int_{[a,b]} \vert f(x)\vert^p \,\mathrm{d}x\right)^\frac{1}{p}$$ $$d(f,g) = \left(\int_{[a,b]} \vert f(x)-g(x)\vert^p \,\mathrm{d}x\right)^\frac{1}{p}$$ ✓ ✓