Hahn—Banach Theorem

Let $X$ be a normed space, $Y \subset X$ a subspace of finite codimension, and $f$ a linear continuous functional on $Y$. Prove that any extension of $f$ to a linear functional on $X$ is a continuous functional.

Let the Banach space $X$ be:

Let $X$ be a normed space. Prove that the Hahn-Banach extension of any linear continuous functional on $X$ is unique if and only if the dual space $X^{*}$ is strictly normed (see Definition 2.4).

Definition 2.4: A normed space $X$ is called strictly normed (or strictly convex) if for $\left\| x\right\| = \left\| y\right\| = 1$ the equality $\left\| x+y\right\| = \left\| x\right\| +\left\| y\right\|$ holds if and only if $x = y$ (in other words, the unit sphere of $X$ contains no line segments).

Prove that any linear continuous functional, defined on an everywhere dense linear subspace of a normed space $X$, has a unique Hahn-Banach extension.

Prove that for any nonzero $x \in X$ there exists a functional $f \in X^{*}$ such that $\left\| f\right\| = 1, f(x) = \left\| x\right\|$.

Prove that for any $x, y \in X, x \neq y$, there exists a functional $f \in X^{*}$ such that $f(x) \neq f(y)$.

Let $x_{1}, x_{2}, \ldots , x_{n}$ be a linearly independent system of vectors of a normed space $X$, and let $\lambda_{1}, \lambda_{2}, \ldots , \lambda_{n}$ be complex numbers. Prove that there exists a linear continuous functional $f$ such that $f\left(x_{k}\right) = \lambda_{k}$ for all $1 \leqslant k \leqslant n$ (compare with problem 5.4).

Let $X$ be a real normed space, $\left\{ x_{n}\right\}$ a finite or countable system of vectors in $X$, and $\left\{ c_{n}\right\} \subset \mathbb {R}$. Prove that the existence of a functional $f \in X^{*}$ with the property $f\left(x_{n}\right) = c_{n}$ for every $n \in \mathbb {N}$ is equivalent to the condition that for an arbitrary finite collection $\left\{ \lambda_{n}\right\}_{n = 1}^{N} \subset \mathbb {R}$ we have $\left|\sum_{n = 1}^{N} \lambda_{n} c_{n}\right| \leqslant M\left\| \sum_{n = 1}^{N} \lambda_{n} x_{n}\right\|$ with some single constant $M > 0$.

Prove that there exists a linear continuous functional $f$ on the real space $l_{\infty }$ satisfying the following conditions:

1. $\inf_{n \in \mathbb {N}} x_{n} \leqslant f(x) \leqslant \sup_{n \in \mathbb {N}} x_{n}$;

2. if $\lim_{n \rightarrow \infty } x_{n} = a$ exists, then $f(x) = a$;

3. the functional is invariant under shifts, i.e., $f\left(\left(x_{1}, x_{2}, x_{3}, \ldots \right)\right) = f\left(\left(x_{2}, x_{3}, \ldots \right)\right)$ for every $x = \left(x_{1}, x_{2}, \ldots \right) \in l_{\infty }$.

Find:

Give examples of a smooth and a non-smooth space. Prove that in a smooth space the Hahn-Banach extension of any linear functional from any one-dimensional subspace is unique.

Let a linear functional $f$ be defined on a one-dimensional subspace $\operatorname {Lin}(x) \subset C[0,1]$, where:

Let a functional $f(\lambda x) = \lambda x\left(t_{0}\right)$ be defined on a one-dimensional subspace $\operatorname {Lin}(x) \subset C[0,1]$, generated by some nonzero function $x(t)$, where $t_{0} \in [0,1]$ is a fixed point. Is the Hahn-Banach extension of such a functional unique?

Prove that for a linear subspace $Y$ of a normed space $X$ with $\bar{Y} \neq X$ and an arbitrary point $x \in X$, there exists a functional $f \in X^{*}$ such that $\left\| f\right\| = 1, f(x) = \operatorname {dist}(x, Y)$, and $f(y) = 0$ for every $y \in Y$.

Let $U_{1}$ and $U_{2}$ be two convex nonempty disjoint sets of a real Banach space $X$, one of which has nonempty interior. Prove that there exist a linear continuous functional $f$ and a number $c \in \mathbb {R}$ such that the affine hyperplane $X_{f, c}$ separates $U_{1}$ and $U_{2}$.

Let $P$ be the subspace of the real space $C[0,1]$ consisting of all polynomials. Prove that $P_{+}$ and $P_{-}$, the sets consisting of polynomials with positive (negative) leading coefficient, are convex, disjoint, but not separated by any hyperplane.

Let $B_{1}$ and $B_{2}$ be two convex closed nonempty disjoint sets of a real Banach space $X$, one of which is compact. Prove that there exist a linear continuous functional $f$ and a number $c \in \mathbb {R}$ such that the affine hyperplane $X_{f, c}$ strictly separates $B_{1}$ and $B_{2}$.