Show that $(C[0,1],\|\cdot\|_{L^1})$ is not a Banach space
7345
Is $C[0,1]$ a Banach space for the $L^1$ norm?
67383
Prove that $G'$ is not planar.
14516
The space of continuous functions $C([0,1])$ is not complete in the $L^2$ norm
128766
Show that the set of functions $\mathbb{N}\to\{0,1\}$ is not countable
80735
Show that $C_0([a, b], \mathbb{R})$ is not $\sigma$-compact
132491
How to show that $C=C[0,1]$ is a Banach space
92497
Isometric Embedding of a separable Banach Space into $\ell^{\infty}$
82355
When is the image of a linear operator (between Banach spaces) closed?
74501
$\delta : C([0,1])\to\mathbb{R}$, $f\mapsto f(0)$, with sup-norm is bounded but unbounded with one-norm.
27775
Proving a function is not a contraction
26183
Prob. 9, Sec. 4.3 in Kreyszig's Functional Analysis Book: Proof of the Hahn Banach Theorem without Zorn's Lemma
26989
Show that C[0,1] is a vector space
34173
Banach spaces and their unit sphere
46326
The space $C[a,b]$ with $\|f\|:=\int_a^b |f(x)|d(x)$ is a normed space but not a Banach space
115272
Showing that projections $\mathbb{R}^2 \to \mathbb{R}$ are not closed
139272
On the norm of a quotient of a Banach space.
43535
show that the interval of the form $[0,a)$ or $(a, 1]$ is open set in metric subspace $[0,1]$ but not open in $\mathbb R^1$