Show that this metric space is incomplete: $C[0,1]$ with metric defined as follows: $d(f,g) = \int_0^1 |f(x)-g(x)| \mathrm{d}x$
67625
$C([0, 1])$ is not complete with respect to the norm $\lVert f\rVert _1 = \int_0^1 \lvert f (x) \rvert \,dx$
15365
Proving that the triangle inequality holds for a metric on $\mathbb{C}$
9355
If $d(x,y)$ is a metric, then $\frac{d(x,y)}{1 + d(x,y)}$ is also a metric
35291
How to show $d(x,y)= \sqrt{d_1(x_1,y_1)^2+d_2(x_2,y_2)^2}$ is a metric?
69528
Metrizability of quotient spaces of metric spaces
17067
To prove that $(\mathbb{P(N)},d)$ is a metric space
154978
Some equivalent formulations of compactness of a metric space
107321
How to show convexity of a ball in metric space?
51227
Proving that given metric space is complete: $X := (0,\infty)$ and $d:=|\ln(x)-\ln(y)|$
64260
Prove that a function between metric spaces is continuous iff the inverse image of a closed set is closed
123753
In a metric $(X,d)$, prove that for each subset $A$, $x\in\bar{A}$ if and only if $d(x,A)=0.$
83159
Defining a metric space
92005
Existence of compact/complete metric on countable set
146485
Prove that the product space is a metric space.
33859
How to show that a disjoint sum of metric spaces is metrizable?
114061
Construct such $d$ that $(\mathbb{R} \setminus \mathbb{Z}, d)$ is complete metric space
123061
$X$ metric separable then $C(X)$ separable