A normed space is Banach iff its unit sphere is complete
34173
Banach spaces and their unit sphere
73462
Nested sequences of balls in a Banach space
107984
Is $C^1[a,b]$ with the norm $\left \| f \right \|_1=(\int_{a}^{b}\left | f(t) \right |dt)+(\int_{a}^{b}\left | f´(t) \right |dt)$ a complete space?
67625
$C([0, 1])$ is not complete with respect to the norm $\lVert f\rVert _1 = \int_0^1 \lvert f (x) \rvert \,dx$
59697
Is the boundary of the unit sphere in every normed vector space compact?
149664
Norm is sup of inner products (proof).
65876
A normed space of continuous functions with norm $\int_{0}^{1}|f(t)|dt$ is not complete
14516
The space of continuous functions $C([0,1])$ is not complete in the $L^2$ norm
52104
A pseudometric on the space of the measurable functions is complete
154762
Dual space of a finite dimensional normed space
97391
Prove that $(C^1[0,1], \|\cdot\|)$ is not a Banach space
49087
Show that a norm on an inner product space satisfies parallelogram law.
129243
Is $C_{c}^{\infty}[a,b]$ a complete metric space?
95887
How to show that quotient space $X/Y$ is complete when $X$ is Banach space, and $Y$ is a closed subspace of $X$?
62428
How to prove this inequality in Banach space?
126744
Is any Banach space a dual space?
30488
Convergence of a sequence implies convergence of the norm of the sequence