Dual Spaces

Prove that if $L$ is an infinite-dimensional normed space, then there is a discontinuous linear functional on it.

Prove that the norm of a functional $f\in L'$ is the reciprocal of the distance from zero to the hyperplane $f(x)=1$.

Prove that any finite-dimensional normed linear space $L$ is reflexive.

Prove that a closed subspace of a reflexive space is reflexive.

Prove that the space $c_0$ of all sequences of real numbers converging to zero, with the norm $p(\left\{ x_n\right\} )=\max \left|x_n\right|$, is not reflexive.

Let $L$ be an infinite-dimensional normed space. Prove that the weak topology in $L$ does not coincide with the strong topology.

Let $L=l_1(\mathbb {R})$ be the space of sequences of real numbers with the norm $p_1(\left\{ x_n\right\} )=\sum_{n=1}^{\infty }\left|x_n\right|$. Prove that weak convergence coincides with strong convergence for sequences in $L$.

A hyperplane $P$ is called a support of a convex set $K$ as it has a common point with $K$ and the whole of $K$ is located on one side of $P$. Prove that it is natural to label the support planes of the unit ball in $L$ with points of the unit sphere in $L'$.

Suppose that the unit ball $B$ of the normed space $L$ is a convex polyhedron. Construct a natural correspondence between the $k$-dimensional faces of $B$ and the $(n-k)$-dimensional faces of the unit ball $B'$ in $L'$.

Let $c$ be the space of all real sequences $\left\{ x_n\right\}$ for which $\lim_{n\to \infty }x_n$ exists, endowed with the norm $\left\| \left\{ x_n\right\} \right\| =\sup \left|x_n\right|$, and let $c_0$ be the subspace of sequences converging to zero. Prove that the spaces $c'$ and $c_0'$ are isomorphic to the space $l_1(\mathbb {R})$, but that the spaces $c$ and $c_0$ are not isomorphic to each other.

Let $L_1$ and $L_2$ be normed spaces whose elements can be expressed as column vectors of lengths $n_1$ and $n_2$, respectively. An operator $A\in \mathcal L(L_1,L_2)$ can then be expressed as a matrix with $n_1$ columns and $n_2$ rows, so that the action of $A$ on an $x\in L_1$ amounts to multiplication of $x$ from the left by $A$ according to the rules of matrix multiplication. Show that the spaces $L_1'$ and $L_2'$ can be identified with the spaces of row vectors of lengths $n_1$ and $n_2$, respectively, so that the action of the adjoint operator $A'$ amounts to multiplication from the right by the matrix of $A$.