(AFr 4.3.1)
Let $Y\subset X$ be a closed linear subspace of a normed
linear space $X$. Then for any $\varepsilon>0$ there exists $z\in X$ s.t.
$\|z\|=1$ and $\|z-y\|>1-\varepsilon$, for any $y\in Y$.
Theorem.
(AFr 4.3.2)
If $Y$ is a finite-dimensional linear subspace of a normed linear
space, then $Y$ is closed.
Theorem.
(AFr 4.3.3)
A normed linear space is of finite dimension iff every bounded
subset is relatively compact.
Example.
Let $X=L^2(0,2\pi)$ and let
$e_k(x)=e^{ikx}/\sqrt{2\pi}$, $k=0, \pm1, \dots$.
Then
$$
\int_0^{2\pi} |e_k(x)|^2dx = 1 \qquad {\text{and}} \qquad
\int_0^{2\pi} |e_k-e_j|^2dx = 2.
$$
The set $\{e_k\}_{k\in{\Bbb Z}}$ is not relatively compact.
Home exercises.
1. (ex. 4.3.4 from AFr)
A norm $\|\cdot\|$ is called strictly convex if $\|x\| = \|y\| =1$ and
$\|x+y\| = 2$ implies $x=y$.
Show that $L^p(\Bbb R^n)$ is strictly convex if $ 1 < p < \infty$ and
is not strictly if $p = 1$ or $p = \infty$.
2. (ex. 4.3.5 from AFr)
Prove that $C[a,b]$ is not equivalent to $L^p$ norm if $1\le p < \infty$.