Normed spaces

Devise measure spaces $(M, \Sigma , \mu )$ for which inequalities (2.1) and (2.2) turn into the numerical Hölder and Minkowski inequalities $(a_{k}, b_{k} \in \mathbb {C})$:

Give examples of measurable functions $x$ such that:

For functions $x \in L_{1}(M, \Sigma , \mu ), y \in L_{\infty }(M, \Sigma , \mu )$ prove the inequality $\int_{M}\left|x(t) y(t)\right| d \mu \leqslant \left\| x\right\|_{1}\left\| y\right\|_{\infty }$.

Prove that if $x \in L_{\infty }(M, \Sigma , \mu )$ and the measure $\mu$ is finite, then $\lim_{p \rightarrow \infty }\left\| x\right\|_{p} = \left\| x\right\|_{\infty }$. In particular, for any vector $x \in \mathbb {C}^{n}$ it holds that

Verify the axioms of a norm in the spaces (see the list of spaces):

Prove that the norms $\left\| \cdot \right\|_{1}$ and $\left\| \cdot \right\|_{2}$, introduced on the space $C^{n}[a, b]$ in the list of spaces, satisfy the axioms of a norm and are equivalent.

Among the mappings of the linear space of continuously differentiable functions on the segment [ 0,1 ] listed below, choose the mappings satisfying the axioms of a norm, and order these norms with respect to the subordination relation:

Let the function $v(t)$ be continuous and positive on $[0,1]$. Prove that the mapping $F: x \mapsto \left(\int_{0}^{1} v(t)|x(t)|^{p} d t\right)^{1 / p}$, acting from $L_{p}[0,1]$ to $\mathbb {R}$, where $p \in [1, \infty )$, satisfies the axioms of a norm, and this norm is equivalent to the standard $\left\| \cdot \right\|_{p}$.

Verify the axioms of a norm on the space of functions of bounded variation $B V[0,1]$.

Prove that all norms $\left\| x\right\|_{a} = \operatorname {Var}_{0}^{1} x(t)+\left|x(a)\right|$, where $0 \leqslant a \leqslant 1$, on the space of functions of bounded variation $B V[0,1]$ are equivalent to each other and equivalent to the norm $\left\| \cdot \right\|_{B V}$ from the list of spaces.

Verify the axioms of a norm in the space $B V_{0}[0,1]$ (see the list of spaces).

Prove Theorem 2.1 on the equivalence of norms in a finite-dimensional space.

Theorem 2.1: If $\dim X < \infty$ (here $\dim X$ is the linear dimension of the linear space $X$) and $\left\| \cdot \right\|_{1}$ and $\left\| \cdot \right\|_{2}$ are two different norms on $X$, then $\left\| \cdot \right\|_{1} \sim \left\| \cdot \right\|_{2}$, i.e., there exist constants $c, C > 0$ such that $c\left\| x\right\|_{1} \leqslant \left\| x\right\|_{2} \leqslant C\left\| x\right\|_{1}$ for all $x \in X$.

Let $p$ be a numerical function on a linear space $X$ satisfying the first two axioms of a norm. Prove that the third axiom (the triangle inequality) is equivalent to the convexity of the set $\left\{ x \in X: p(x) \leqslant 1\right\}$.

Prove that the function $\left\| x\right\|_{p} = \left(\sum_{k = 1}^{n}\left|x_{k}\right|^{p}\right)^{1 / p}$ is not a norm for $0 < p < 1$ on an $n$-dimensional linear space for $n \geqslant 2$. $p \leqslant q$.

Find $\inf_{x \neq 0} \frac{\left\| x\right\|_{p}}{\left\| x\right\|_{q}}$ and $\sup_{x \neq 0} \frac{\left\| x\right\|_{p}}{\left\| x\right\|_{q}}$, where $x \in \mathbb {R}^{n}$, for all $p$ and $q \in [1, \infty ]$,

Prove that in the linear space $l_{p}, 1 \leqslant p \leqslant \infty$, the norms $\left\| \cdot \right\|_{p}$ and $\left\| \cdot \right\|_{q}, q > p$, are not equivalent, but $\left\| \cdot \right\|_{q} \prec \left\| \cdot \right\|_{p}$.

Prove that the norms $\left\| \cdot \right\|_{p}$ and $\left\| \cdot \right\|_{q}, 1 \leqslant p < q \leqslant \infty$, in the linear space of continuous functions on $[0,1]$ are not equivalent, but $\left\| \cdot \right\|_{p} \prec \left\| \cdot \right\|_{q}$.

Find necessary and sufficient conditions on a sequence $\left\{ a_{n}\right\}_{1}^{\infty }$, where all $a_{n} > 0$, under which the following sets are bounded subsets of $l_{p}, p \in [1, \infty )$:

Find necessary and sufficient conditions on a sequence $\left\{ a_{n}\right\}$, where all $a_{n} > 0$, under which the following sets are open subsets of $l_{p}, p \in [1, \infty )$:

Prove that in any normed space there exist two disjoint open sets that cannot be placed into two disjoint closed sets. Do such sets exist in every metric space?

Let $M$ be a subset of $\mathbb {C}$. Consider the set $\mathbf{M} = \left\{ x \in C[0,1]: x(t) \in M\text{ for all }t \in [0,1]\right\}$. Prove that the set М is open (closed) if and only if $M$ is open (respectively, closed).

In the ball $B(0,1)$ of the space $l_{2}$, place a countable number of disjoint balls of radius $\frac{1}{10}$.

Give examples of sequences $\left\{ x_{n}\right\}_{1}^{\infty }$ with the following properties:

Let two norms $\left\| \cdot \right\|_{1}$ and $\left\| \cdot \right\|_{2}$ be given on a linear space $X$. Prove that the following properties are equivalent:

1. $\left\| \cdot \right\|_{1} \prec \left\| \cdot \right\|_{2}$;

2. any sequence $\left\{ x_{n}\right\}_{n = 1}^{\infty }$, converging in the normed space $\left(X,\left\| \cdot \right\|_{2}\right)$ to $x$, converges to $x$ also in $\left(X,\left\| \cdot \right\|_{1}\right)$;

3. any sequence $\left\{ x_{n}\right\}_{n = 1}^{\infty }$, fundamental in the normed space $\left(X,\left\| \cdot \right\|_{2}\right)$, is also fundamental in $\left(X,\left\| \cdot \right\|_{1}\right)$;

4. the closure of an arbitrary set $A \subset X$ with respect to the first norm contains the closure of this set with respect to the second norm;

5. the mapping $A: x \mapsto x$ from $(X,\left\| \cdot \right\|_{2})$ to $(X,\left\| \cdot \right\|_{1})$ is continuous. In particular, if $\left\| \cdot \right\|_{1} \sim \left\| \cdot \right\|_{2}$, then the spaces $X_{1} = \left(X,\left\| \cdot \right\|_{1}\right)$ and $X_{2} = \left(X,\left\| \cdot \right\|_{2}\right)$ are isomorphic.

Prove that $c_{0}$ is a closed subspace of $c$, and $c$ is a closed subspace of $l_{\infty }$. Prove that $c_{00}$ is a non-closed linear subspace of the space $c$ and of the space $c_{0}$. Find the closure of this subspace.

Prove that in the space $l_{p}$ the linear subspace $l_{q} = \left\{ x \in l_{p}: \sum_{n = 1}^{\infty }\left|x_{n}\right|^{q} < \infty \right\}$, where $1 \leqslant q < p \leqslant \infty$, is not a closed subspace. Find its closure.

Prove that in a normed space a ball cannot contain a nonzero linear subspace, and a linear subspace not coinciding with the whole space cannot contain any ball.

Prove that for any set $M$ in a normed space $\operatorname {Lin}(\bar{M}) \subset \overline{\operatorname {Lin}(M)}$. Prove that for $M = \left\{ e_{n}\right\}_{1}^{\infty }$ in $l_{1}$, where $e_{n} = \left(0, \ldots , 0, \frac{1}{n}, 0,0, \ldots \right)$, these sets are different.

Prove that in a normed space $X$ any finite-dimensional subspace is closed.

Do the following sets of continuous functions form closed subspaces in the space $C[-1,1]$ or $C_{2}[-1,1]$:

Do the following sets of continuous functions form closed subspaces in the space $B C(\mathbb {R})$:

Is the set $M = \left\{ x \in l_{p}: \sum_{n = 1}^{\infty } x_{n} = 0\right\}$ a closed linear subspace of the space $l_{p}, p \in [1, \infty ]$?

Prove that if some closed subspace of $C[0,1]$ consists of continuously differentiable functions, then it is finite-dimensional.

Prove that if some closed subspace of $L_{1}[0,1]$ consists of continuous functions, then it is finite-dimensional.

Give an example of a cone $K$ in a normed space $X$ such that $K$ has empty interior and $\operatorname {Lin}(K)$ is everywhere dense in $X$. Show that if $\operatorname {dim} X < \infty$, then such an example is impossible.

Which of the spaces listed below are strictly normed:

Prove that any uniformly convex normed space is strictly normed. Deduce from this that the spaces $l_{1}, l_{\infty }, c, c_{0}, L_{1}[0,1], L_{\infty }[0,1], C[0,1]$ and $B V[0,1]$ are not uniformly convex.

Prove that the spaces $l_{p}$ and $L_{p}, p \in (1, \infty )$, are uniformly convex.

Prove that the Minkowski functional $p_{A}$ of a convex, balanced, absorbing set $A$ in a linear space $X$ is a seminorm. Prove, conversely, that for any seminorm $p$ on $X$ the set $M = \left\{ x \in X: p(x) \leqslant 1\right\}$ is convex, balanced, and absorbing, and its Minkowski functional $p_{M} = p$.

Let $A$ be a convex, balanced, and absorbing set in a linear space $X$. Prove that its Minkowski functional $p_{A}$ is a norm if and only if the set $A$ contains no line.

In the linear space $\mathbb {R}^{2}$ find sets whose Minkowski functionals coincide with the norms $\left\| \cdot \right\|_{p}$, where $p \in [1, \infty ]$.

Prove that a regular $2n$-gon centered at zero is the unit ball for some norm on the space $\mathbb {R}^{2}$. Prove that a regular $(2n+1)$-gon centered at zero is not the unit ball for any norm.

Let $Y$ be a linear subspace of a normed space $(X,\left\| \cdot \right\| )$. Prove that any norm on $Y$ equivalent to $\left\| \cdot \right\|$ can be extended to all of $X$ preserving the axioms of a norm and the equivalence property.

Let the norms $\left\| \cdot \right\|_{1}$ and $\left\| \cdot \right\|_{2}$ on a linear space $X$ be equivalent. Prove that the normed space $(X,\left\| \cdot \right\|_{1})$ is complete if and only if the normed space $\left(X,\left\| \cdot \right\|_{2}\right)$ is complete. ¹ Prove that if $\left\| \cdot \right\|_{1} \sim \left\| \cdot \right\|_{2}$ on $X$, then this relation is preserved on the completed space as well.

Let the normed spaces $X$ and $Y$ be isomorphic. Prove that then they are either both complete or both incomplete.

Prove that any finite-dimensional normed space is a Banach space.

Prove the completeness of the following spaces:

Prove that the space $c_{00}$ is not complete, and find its completion (compare with problem 2.26).

Prove that the spaces $C_{p}[0,1], p \in [1, \infty )$, are not complete.

Prove that $L_{p}[0,1]$ is the completion of $C_{p}[0,1]$ for $p \in [1, \infty )$, but $L_{\infty }[0,1]$ is not the completion of the space $C[0,1]$.

Prove that the spaces $C_{p}^{n}[0,1], p \in [1, \infty ), n \in \mathbb {N}$, are not complete.

Prove that the space $A C[0,1]$ is the completion of the space $C_{1}^{1}[0,1]$ and, by Theorem 2.4, is isomorphic to the space $W_{1}^{1}[0,1]$.

Theorem 2.4: Every normed space has a completion. If $Y_{1}$ and $Y_{2}$ are completions of the same space $X$, then they are isometrically isomorphic.

Prove that for any function $x \in W_{1}^{1}$ the equality $\left\| x\right\|_{W_{1}^{1}} = \operatorname {Var}_{0}^{1} x+\int_{0}^{1}\left|x(t)\right| d t$ holds.

Prove that for any continuously differentiable function $x(t)$ on the segment $[0,1]$ the inequality holds

for every $p \in [1, \infty )$. Using this, prove the embedding theorem (G. H. Hardy, J. E. Littlewood, 1928): the space $W_{p}^{1}[0,1]$ is embedded in the space $C[0,1]$.

Prove the following statements:

Which of the following functions belong to the space $W_{2}^{1}[-1,1]: x_{1}(t) = t, x_{2}(t) = \operatorname {sign} t, x_{3}(t) = \left|t\right|, x_{4}(t) = \left|t\right|^{a}$, where $a \in \mathbb {R}$? For those functions that belong to this space, find the norm.

Prove that the space $\mathcal{P}_{n}[0,1], n \in \mathbb {N}$, is complete, while the space $\mathcal{P}[0,1]$ is not complete. Find the completion of the space $\mathcal{P}[0,1]$.

Find the completion of the linear space of polynomials on the segment $[0,1]$ with respect to the norm of the space $B V[0,1]$.

Prove that on the linear space of polynomials a norm cannot be introduced in such a way as to obtain a Banach space.

Let $X$ be a metric space. Denote by $B C(X)$ the set of all continuous bounded functions from $X$ to $\mathbb {C}$. Prove that the norm $\left\| f\right\|_{B C(X)}: = \sup_{x \in X}\left|f(x)\right|$ turns $B C(X)$ into a Banach space (the linear operations are introduced in the natural way).

Let $X$ be a metric space. Prove that there exists a mapping $F: X \rightarrow B C(X)$ such that $\left\| F(x)-F(y)\right\|_{B C(X)} = \rho_{X}(x, y)$ for any $x, y \in X$.

Prove that every finite metric space $X$ embeds isometrically in the space $l_{\infty }(n)$ for some $n$.

Using the previous problem, prove Theorem 1.1.

Theorem 1.1 (on completion): Every metric space has a completion. If $Y_{1}$ and $Y_{2}$ are completions of the same space $X$, then they are isometric.

Here (Definition 1.16) a metric space $Y$ is called a completion of a metric space $X$ if: (1) $Y$ is complete; (2) there is an isometric embedding $f: X \rightarrow Y$ such that the set $f(X)$ is everywhere dense in $Y$.

Prove that the following spaces embed isometrically in $B V_{0}[0,1]$:

Prove that in a finite-dimensional normed space any bounded sequence of vectors has at least one limit point.

Determine which of the following sequences converge in the space $C[0,1]$, and find their limits. For sequences that do not converge, find all limit points.

Prove that in a Banach space any system of nested closed balls has nonempty intersection ¹.

Prove that a normed space $X$ is complete if and only if the Weierstrass criterion for convergence of series holds in it: any absolutely convergent series converges. Give an example of a series that converges to nothing, but is absolutely convergent, in an incomplete normed space.

Let $X$ be a Banach space. Prove that any finite-dimensional subspace in $X$ is a subspace of existence.

Give an example of a Banach space $X$, a closed linear subspace $Y$ in it, and a vector $x$ for which the element of best approximation in the subspace $Y$ is not unique.

Prove that in a normed space $X$ the element of best approximation is unique for every closed linear subspace $Y \subset X$ if and only if the 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).

(P. L. Chebyshev, 1859). Prove that among all polynomials of the form $t^{n}+a_{n-1} t^{n-1}+\ldots +a_{0}$, the smallest norm in the real space $C[-1,1]$ is attained by the Chebyshev polynomial (of the first kind) $T_{n}(t) = 2^{1-n} \cos (n \arccos t)$.

Let $X$ be a normed space, and $X_{0} \varsubsetneqq X$ its closed subspace. Prove that for any $\varepsilon > 0$ there exists an $\varepsilon$-perpendicular to $X_{0}$.

Let $X_{0}$ be a closed subspace of a Banach space $X$. Prove that if for some vector $x \notin X_{0}$ there exists an element of best approximation in the subspace $X_{0}$, then there exists a $0$-perpendicular to the subspace $X_{0}$.

Prove that for every finite-dimensional subspace of a Banach space there exists a $0$-perpendicular.

Let the linear space $X = X_{1} \times X_{2}$, where $(X_{1},\left\| \cdot \right\|_{1})$ and $(X_{2},\left\| \cdot \right\|_{2})$ are normed spaces. Verify the axioms of the norm $\left\| \binom {x_{1}}{x_{2}}\right\|_{X} = \left\| x_{1}\right\|_{X_{1}}+\left\| x_{2}\right\|_{X_{2}}$ in the space $X$. Prove that if $X_{1}$ and $X_{2}$ are Banach spaces, then the space $X$ is also a Banach space.

Prove that the Cartesian product $c_{0} \times c_{0}$ with the norm $\left\| \left(x_{1}, x_{2}\right)\right\| = \left\| x_{1}\right\|_{c_{0}}+\left\| x_{2}\right\|_{c_{0}}$ is not isometrically isomorphic to the space $c_{0}$.

Give an example of a norm $\left\| \binom {\cdot }{\cdot }\right\|$ on the linear space $\mathbb {R}^{2}$ and normed spaces $\left(X_{1},\left\| \cdot \right\|_{1}\right),\left(X_{2},\left\| \cdot \right\|_{2}\right)$ such that the function $p\left(x_{1}, x_{2}\right) = \left\| \binom {\left\| x_{1}\right\|_{1}}{\left\| x_{2}\right\|_{2}}\right\|$ on the linear space $X_{1} \times X_{2}$ is not a norm. Prove that if the norm $\left\| \binom {\cdot }{\cdot }\right\|$ is monotonic in each coordinate, then for any normed spaces $X_{1}, X_{2}$ the function $p\left(x_{1}, x_{2}\right)$ is a norm on the linear space $X_{1} \times X_{2}$.

Give an example of a norm on the Cartesian product $X_{1} \times X_{2}$ of normed spaces that is not given by the method indicated in problem 2.79.

Let $X_{0}$ be a closed subspace of a normed space $X$. Prove that $\left\| [x]\right\|_{0} = \inf_{x \in [x]}\left\| x\right\|$ defines a norm on the linear space $X / X_{0}$.

Let $X_{0}$ be a closed subspace of a Banach space $X$. Prove that the normed space $X / X_{0}$ is complete.

Let $X_{0}$ be a closed subspace of a normed space $(X,\left\| \cdot \right\| )$. Prove that if the spaces $(X_{0},\left\| \cdot \right\| )$ and $\left(X / X_{0},\left\| \cdot \right\|_{0}\right)$ are complete, then $X$ is complete as well.

Prove that:

Prove Theorem 2.6.

Theorem 2.6: If $X$ is an arbitrary normed space and $X_{1}$ and $X_{2}$ are two of its subspaces, one of which is closed and the other finite-dimensional, then their linear direct sum is topological.

Prove that the linear direct sum of the subspaces $c_{00}$ and $\operatorname {Lin}\left(x_{0}\right)$, where $x_{0} = \left(1, \frac{1}{2}, \frac{1}{3}, \ldots \right)$, in the space $c$ is not topological.