Normed Spaces

Prove that $C(X)$ is a Banach space for any compact space $X$.

Prove that $l_p(K)$ is a separable Banach space for $1\leq p<\infty$ and $K=\mathbb {R}$ or $\mathbb {C}$.

Prove that the space $l_\infty (K)$ of all bounded sequences $\left\{ x_n\right\}$, $x_n\in K$, is a nonseparable Banach space with the norm $\left\| \left\{ x_n\right\} \right\| =\sup_n\left|x_n\right|$.

Prove that a normed space $L$ is a Banach space if and only if every series $\sum x_i$ for which $\sum \left\| x_i\right\| <\infty$ converges in $L$.

Prove that a normed space $L$ is nonseparable if and only if it contains an uncountable set of pairwise disjoint balls of radius $1$.

Prove that in a normed space the distance from a given point to an arbitrary finite-dimensional subspace is attained.

Find the distance from the point $x^n\in C[-1,1]$ to the subspace $P_{n-1}\subset C[-1,1]$ consisting of all the polynomials of degree $<n$.

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 $A$.

Suppose that a Banach space $L$ is decomposed into an algebraic direct sum: $L=L_1+L_2$. Prove that the operator of projection onto $L_1$ parallel to $L_2$ is bounded if and only if $L_1$ and $L_2$ are closed in $L$.

Prove that an operator $P$ in a Banach space $L$ is a projection onto some closed subspace $L_1$ parallel to a closed subspace $L_2$ if and only if it is bounded and satisfies the relation $P^2=P$.

For which functions $a(x)$ is the operator of multiplication by $a(x)$ a continuous operator from $L^{p}\left({[0,1]}\right)$ into $L^{q}\left({[0,1]}\right)$?

Let $A(t)$ be a differentiable operator-valued function on $\mathbb {R}$ with values in $\operatorname {End}L$, $\dim L<\infty$. Prove that all the solutions of the differential equation $A'(t)=CA(t)$, where $C\in \operatorname {End}L$, have the form $A(t)=e^{tC}A_0$, where $A_0\in \operatorname {End}L$ and $e^{tC}=\sum_{k=0}^{\infty }(t^kC^k/k!)$.

Prove that the assertion of first-edition Problem 363 ceases to be true in the case $\dim L=\infty$.

Let $K(x,y)$ be a continuous function on the unit square in $\mathbb {R}^{2}$, and let the operator $A$ act from $L^{p}\left({[0,1]}\right)$ to $L^{q}\left({[0,1]}\right)$, $1\leq p,q<\infty$, by the formula $(Af)(x)=\int_0^1K(x,y)f(y)\, dy$. Find the adjoint operator $A':L^{q'}\left({[0,1]}\right)\toL^{p'}\left({[0,1]}\right)$ where $p'=p/(p-1)$, $q'=q/(q-1)$.

Let $P:C[0,2]\to C[0,1]$ be the restriction operator. Find the adjoint operator $P':V[0,1]\to V[0,2]$.

Here $V[0,b]$ denotes the space of functions of bounded variation on $[0,b]$ that are left-continuous and vanish at $0$, with the total-variation norm; the duality with $C[0,b]$ is given by the Stieltjes integral $f\mapsto \int_0^b f(t)\, dF(t)$.

Let $L$ be a normed space, $L_0$ a closed subspace of it, and $L_1=L/L_0$, with the norm defined in Problem 256(a). Is it true that $L$ is a Banach space if $L_0$ and $L_1$ are Banach spaces?

Prove that every separable Banach space over the field $K$ is a quotient space of $l_1(K)$.

A subspace $L_0$ of a Banach space $L$ is said to be complemented if there exists a closed subspace $L_1\subset L$ such that $L=L_0\oplus L_1$ (algebraic direct sum). Prove that:

Let $L_1$ and $L_2$ be two linear spaces over a field $K$. We consider the category whose objects are bilinear mappings $\varphi \colon L_1\times L_2\to L$, where $L$ is a linear space (depending on the object $\varphi$). A morphism from an object $\varphi \colon L_1\times L_2\to L$ to an object $\psi \colon L_1\times L_2\to M$ is defined to be a linear mapping $\chi \colon L_1\to M$ for which the diagram

commutes. Prove that the resulting category has a universal repelling object $\pi \colon L_1\times L_2\to L_1\otimes_K L_2$. The space $L_1\otimes_K L_2$ is called the tensor product of the spaces $L_1$ and $L_2$ over the field $K$.

Let $L_1$ and $L_2$ be Banach spaces, and $\mathcal K$ the category of bilinear mappings $A:L_1\times L_2\to L$ with norm $\leq 1$, $L$ a Banach space. The morphisms are defined as in Problem 260, with the extra condition $\left\| \varphi \right\| \leq 1$. Prove that the mapping $A_0:L_1\times L_2\to L_1\widehat\otimes L_2:(x_1,x_2)\mapsto x_1\otimes x_2$ is a universal object in the category $\mathcal K$.

Let $L_1=l_2(n,\mathbb {R})$, $L_2=l_2(m,\mathbb {R})$. The space $L_1\otimes L_2$ can be identified with the space of $n\times m$ matrices as follows. Write the elements of $L_2$ as row vectors, and those of $L_1$ as column vectors. Then the matrix $x\cdot y$ is assigned to the element $x\otimes y$. Suppose that $a\in L_1\otimes L_2$ corresponds to the matrix $A$. Prove that the norm of $a$ in $L_1\widehat\otimes L_2$ can be computed by the formula $\left\| a\right\| =s_1^{1/2}+\cdots +s_k^{1/2}$, $k=\min (n,m)$, where the $s_i$ are the eigenvalues of the matrix $AA'$ ($A' A$), arranged in decreasing order, if $k=n$ (resp., $k=m$).

Prove that under the conditions of Problem 262 the norm of $a\in L_1\underset {\vee }{\otimes }L_2$ coincides with the norm of the matrix $A$ as an operator from $l_2(m,\mathbb {R})$ to $l_2(n,\mathbb {R})$ and is equal to the number $s_1^{1/2}$.

Let $L_1$ and $L_2$ be Banach spaces. Show that there is a natural isometric imbedding of $L_1'\underset {\vee }{\otimes }L_2'$ into the space $\mathcal L(L_1,L_2)$.

Prove that $l_1(n,\mathbb {R})\widehat\otimes l_1(m,\mathbb {R})$ is isomorphic to $l_1(mn,\mathbb {R})$.

Prove that $l_\infty (n,\mathbb {R})\underset {\vee }{\otimes }l_\infty (m,\mathbb {R})$ is isomorphic to $l_\infty (mn,\mathbb {R})$.