Lecture 7

Ch. 4.8 from Avner Friedman

Theorem. (AFr Th 4.8.2.) (Hahn-Banach )

Let $X$ be a normed linear vector space and let $Y\subset X$ be a linear subspace. Then for any $ y^*\in Y^*$ there exists $x^*\in X^*$ s.t.
$$
\|x^*\| = \|y^*\| \qquad \& \qquad x^*(y) = y^*(y), \quad \forall y\in Y.
$$

Theorem. (AFr Th 4.8.3.)

Let $X$ be a normed linear vector space and let $Y\subset X$ be a linear subspace. Let $x_0\in X$ s.t.
$$
\inf_{y\in Y} \|y-x_0\| = d >0.
$$
Then there exists $x^*\in X^*$ s.t.
$$
x^*(x_0) = 1, \qquad \|x^*\| = \frac{1}{d}\quad {\rm and} \quad x^*(y) = 0, \quad \forall y\in Y.
$$

Corollary. (AFr Th 4.8.4.)

If $X$ is a normed linear space, then for any $x\not= 0$ there exists $x^*\in X^*$ s.t. $\|x^*\| = 1$ and $x^*(x) = \|x\|$.

Corollary. (AFr Th 4.8.5.)

If $X$ is a normed linear space and if $y\not= z$, $y,z\in X$, then there exists $x^*\in X^*$ s.t. $x^*(y)\not= x^*(z)$.

Corollary. (AFr Th 4.8.6.)

Let $X$ be a normed linear vector space. Then for any $x\in X$
$$
\|x\| = \sup_{x^*\not=0} \frac{|x^*(x)|}{\|x^*\|} = \sup_{\|x^*\|=1} |x^*(x)|.
$$

Corollary. (AFr Th 4.8.7.)

Let $X$ be a normed linear vector space and let $Y\subset X$ be a linear subspace. Assume that $\overline{Y} \not=X$. Then there exists $x^*\not= 0$ s.t. $x^*(y) = 0$, $\forall y\in Y$.

Definition.

The null space of $x^*\in X^*$ is the set
$$
N_{x^*} = \{x\in X:\, x^*(x) = 0\}.
$$
Let $x^*\not=0$. Then there is $x_0\not=0$ s.t. $x^*(x)=1$ and any $x\in X$ can be written as $x=z+\lambda x_0$, where $\lambda = x^*(x)$ and $z=x-\lambda x_0 \in N_{x^*}$.

Example.

Let $f\in L^p(0,1)$ $g\in L^q(0,1)$ $1/p+1/q =1 $ be real functions and define a linear functional $G^*$ on $L^p(0,1)$ such that
$$
G^*(f) = \int_0^1 f(x) g(x) \,dx.
$$
Then $N_g = \{f\in L^p(0,1):\, \int f(x) g(x) \,dx = 0\}$.

Definition.

Let $c\in {\Bbb R}$, $ x^*\not=0$. The set
$$
\{x\in X: \, {\rm Re} \,x^*(x) = c\}
$$
is called hyperplane.

The sets $ \{x\in X: \, {\rm Re}\, x^*(x) \ge c\}$, $ \{x\in X: \, {\rm Re}\, x^*(x) \le c\}$ are called half-spaces.

Corollary. (AFr 4.8.8)

Let $X$ be a normed linear space and let $x_0\in \overline{B} = \{x:\, \|x\|\le 1\}$. Then for any $x_0$ s.t. $\|x_0\| = 1$ there is a tangent hyperplane to $B$ at $x_0$. Home exercises.
1. Define $C_0$ as a set of sequences $\{a_k\}_{k=1}^\infty$ for which $\lim_{k\to\infty} a_k = 0$. If we introduce the following norm
$$
\|\{a_k\}_{k=1}^\infty\| = \max_{k\in{\Bbb N}} |a_k|,
$$
then $C_0$ becomes a normed linear space.

Assume that $\{\lambda\}_{k=1}^\infty \in l_1$ $\Big(\Leftrightarrow$ $\sum_{k=1}^\infty |\lambda_k| \le \infty\Big)$.
Show that $$ \Lambda(\{a_k\}_{k=1}^\infty) = \sum_{k=1}^\infty \lambda_k a_k $$ is a linear functional on $C_0$ and $\|\Lambda \| = \sum_{k=1}^\infty |\lambda_k|$.

2.
Show that $l_\infty = l_1^*$ but $l_\infty^* \not= l_1$.

3.
Let $X$ be a Banach space. We say that the functional $\varphi$ is convex if
$$
\varphi\Big(\frac{x + y}{2}\Big) \le \frac{1}{2}\, \Big( \varphi(x) + \varphi(y)\Big).
$$
Define a mapping $L:\, X\to X^*$, s.t.
$$
L\varphi (x^*) = \psi(x^*) = \sup_{x\in X} \Big(x^*(x) - \varphi(x)\Big).
$$
Show that $\psi$ is convex.
If now $L^*:\, X^*\to X$ is defined by
$$
L^*\varphi (x) = \sup_{x^*\in X^*} \Big(x^*(x) - \psi(x^*)\Big),
$$
then $L^*L\varphi = \varphi$.