Normed and Seminormed Spaces

Prove the stated assertion. Under the usual operations on numbers, the set of all real (complex) numbers forms a real (complex) linear space.

Prove the stated assertion. The finite-dimensional vector space $R^{n}$, under the usual operations on vectors $\mathbf{x} = \left(x_{1} ; \ldots ; x_{n}\right), x_{k} \in R, k = 1,2, \ldots , n$, forms a real vector space.

Prove the stated assertion. Let $C^{n}$ be the set of all ordered sets $z = \left(z_{1} ; \ldots ; z_{n}\right)$ of $n$ complex numbers $z_{k} \in C, k = 1,2, \ldots , n$, with the linear operation defined as follows: if

then

Then $C^{n}$ is a complex linear space.

Prove the stated assertion. Let $E$ be an arbitrary set. Then the collection $F(E)$ of all functions $f: E \rightarrow R$ (respectively $f: E \rightarrow C$), under the usual definition of addition of functions and multiplication of them by a real (complex) number, is a real (complex) linear space.

Prove the stated assertion. The set $\mathcal{P}^{(n)}$ (respectively $\mathcal{P}_{C}^{(n)}$) of all polynomials of degree not exceeding a given natural number $n$, in one variable, with real (complex) coefficients, is a linear space.

Prove the stated assertion. The set $\mathcal{P} = \bigcup_{n = 0}^{\infty } \mathcal{P}^{(n)}$ (respectively $\mathcal{P}_{C} = \bigcup_{n = 0}^{\infty } \mathcal{P}_{C}^{(n)}$) of all polynomials of one variable with real (complex) coefficients (see problem 5) is a real (complex) linear space.

Prove the stated assertion. The set of all numerical sequences $\left(x_{1} ; \ldots ; x_{n} ; \ldots \right), x_{n} \in R$ (respectively $x_{n} \in C$), n = 1,2, , under the usual definition of their addition and multiplication by a number, is a linear space.

Prove the stated assertion. The set $C(X)$ of all continuous numerical functions $f: X \rightarrow R$ (respectively $f: X \rightarrow C$) on a metric space $X$ is a subspace of the space $F(E)$ of all numerical functions defined on the space $X$ (see problem 4).

Prove the stated assertion. The set $C B(X)$ of all bounded continuous functions $f: X \rightarrow R$ (respectively $f: X \rightarrow C$) on a metric space $X$ is a subspace of the spaces $B(X)$ (see problem 8) and $C(X)$.

Prove the stated assertion. The set $R L_{p}[a ; b]$ of all numerical functions $x(t)$, defined on the segment $[a ; b]$, for which the integral $\int_{a}^{b}\left|x(t)\right|^{p} d t, p > 0$, is finite, is a linear space.

Remark. According to the remark made above, the integral is understood as an improper integral, in particular the Riemann integral. Hence here $R \mathscr {L}_{p}[a, b]$ denotes a broader set of functions than in example 2.

Prove the stated assertion. The set $C^{1}[a ; b]$ of all continuously differentiable functions on the segment $[a ; b]$ forms a linear space that is a subspace of the space $C[a ; b]$ (see problem 8).

Prove the stated assertion. The set $l_{p}$ of all sequences of real numbers $\left(x_{1} ; \ldots ; x_{n} ; \ldots \right)$ for which the series $\sum_{n = 1}^{\infty }\left|x_{n}\right|^{p}$ converges forms a linear space.

Prove the stated assertion. The set of all sequences of complex numbers $\left(x_{1} ; \ldots ; x_{n} ; \ldots \right)$ for which the series $\sum_{n = 1}^{\infty }\left|x_{n}\right|^{p}$ converges forms a linear space.

Prove the stated assertion. If $Y$ and $Z$ are subspaces of a linear space $X$, then $Y+Z$ is also a subspace of this space.

Prove the stated assertion. Vectors $x_{1}, \ldots , x_{n}$ are linearly dependent if and only if at least one of them is a linear combination of the others.

Prove the stated assertion. A system of vectors containing the zero vector is linearly dependent.

Prove the stated assertion. Every subsystem of a linearly independent system is linearly independent.

Prove the stated assertion. If a system of vectors is linearly dependent, then any system containing it is also linearly dependent.

Prove the stated assertion. In a linear $n$-dimensional space there do not exist more than $n$ linearly independent vectors.

Prove the stated assertion. In a linear $n$-dimensional space every basis consists of $n$ vectors.

Prove the stated assertion. In a linear $n$-dimensional space every system of $n$ linearly independent vectors is a basis of this space.

Prove the stated assertion. The space $\mathcal{P}^{(n)}$ of polynomials of one variable of degree not exceeding $n$ (see problem 5) is $(n+1)$-dimensional, and the monomials $1, t, t^{2}, \ldots , t^{n}$ form a basis in it.

Prove the stated assertion. The space of all polynomials of one variable (see problem 6) is an infinite-dimensional space.

Prove the stated assertion. If the vectors $x_{1}, x_{2}, \ldots , x_{n}$ form a basis in a linear space $X, \quad y_{i} = \sum_{j = 1}^{n} a_{i j} x_{j}, i = 1,2, \ldots , n$, and $\operatorname {det}\left(a_{i j}\right) \neq 0$, then the vectors $y_{1}, y_{2}, \ldots , y_{n}$ also form a basis in the space $X$.

Prove the stated assertion. If the vectors $x_{1}, x_{2}, \ldots , x_{n}$ form a basis in a linear space $X, y_{i} = \sum_{j = 1}^{n} a_{i j} x_{j}$, where $\left(a_{i j}\right)$ is a triangular matrix and $a_{i j} \neq 0, i = 1,2, \ldots , n$, then the vectors $y_{1}, y_{2}, \ldots , y_{n}$ also form a basis in the space $X$.

Prove the stated assertion. Every system of polynomials $P_{m}(t) = \sum_{k = 0}^{m} a_{m k} t^{k}$ of degrees $m = 0,1, \ldots , n$ (i.e., $a_{m m} \neq 0$) forms a basis in the space $\mathcal{P}^{(n)}$ (see problem 5).

Prove the stated assertion. If a system of polynomials is linearly dependent in the linear space of polynomials $\mathcal{P}$ (see problem 6), then it is linearly dependent on any interval as well. If a system of polynomials is linearly dependent on some interval of positive length, then it is linearly dependent in the space $\mathcal{P}$ as well.

Prove the stated assertion. The Legendre polynomials

form a basis in the space $\mathcal{P}^{(n)}$ (see problem 5).

Prove the stated assertion. The function $\varphi_{n}(t) = \cos (n \arccos t), n = 1,2, \ldots ,\left|t\right| \leqslant 1$, is a polynomial of degree $n$. The Chebyshev polynomials

form a basis in the space $\mathcal{P}^{(n)}$ (see problem 5).

Prove the stated assertion. Any polynomial of degree not exceeding $n$ is a linear combination of the Legendre polynomials $P_{0}(t), P_{1}(t), \ldots , P_{n}(t)$ and a linear combination of the Chebyshev polynomials $T_{0}(t), T_{1}(t), \ldots , T_{n}(t)$.

Prove the stated assertion. The kernel of a linear mapping $f: X \rightarrow Y$ is a subspace of the linear space $X$.

Prove the stated assertion. The image of a linear space under a linear mapping into another linear space is a subspace of the latter.

Prove the stated assertion. For a linear mapping $f: X \rightarrow Y$ of a linear space $X$ into a linear space $Y$ to be an injection, it is necessary and sufficient that the kernel of this mapping consist only of zero.

Prove the stated assertion. All linear $n$-dimensional spaces are isomorphic to one another.

Prove the stated assertion. The Fourier transform of continuous, absolutely integrable functions on the whole real line $R$ is an injection into the linear space of functions $f: R \rightarrow C$.

Prove the stated assertion. The set $L(X ; Y)$ of all linear operators mapping a linear space $X$ into a linear space $Y$, under the usual definition of linear operations on them, forms a linear space.

Prove the stated assertion. $R^{n+m} = R^{n} \times R^{m}, n, m = 0,1, \ldots$

Prove that linear functionals $f_{1}$ and $f_{2}$ of a linear space $X$ are linearly dependent if and only if $\operatorname {ker} f_{1} = \operatorname {ker} f_{2}$.

Prove that for a linear functional $f$ of a linear space $X$ to be a linear combination of linear functionals $f_{1}$ and $f_{2}$ of the same space, it is necessary and sufficient that $\operatorname {ker} f_{1} \cap \operatorname {ker} f_{2} \subset \operatorname {ker} f$.

A subspace $H$ of a real (respectively, complex) linear space $X$ is called a hyperplane of the space $X$ if there exists a point $a \in X \backslash H$ such that $X = H \oplus R a$ (respectively $X = H \oplus C a$). Here

Prove that if $H$ is a hyperplane of a real linear space $X$ and $b \in X \backslash H$, then $X = H \oplus R b$.

Prove that if $H_{1}$ and $H_{2}$ are hyperplanes of a linear space $X$ and $H_{1} \subset H_{2}$, then $H_{1} = H_{2}$.

Prove that if $H_{1}$ and $H_{2}$ are hyperplanes of a linear space $X$ and $H_{1} \neq H_{2}$, then there exist points $h_{1} \in X$ and $h_{2} \in X$ such that $h_{1} \in H_{1} \backslash H_{2}, h_{2} \in H_{2} \backslash H_{1}$.

Prove that a subspace $H$ of a linear space $X$ is a hyperplane of this space if and only if $H$ is the kernel of some linear functional.

Prove: if $f_{1}$ and $f_{2}$ are linearly independent linear functionals of a linear space $X$, then there exists a linear functional $g$ of the space $\operatorname {ker} f_{1}$ such that $\operatorname {ker} g = \operatorname {ker} f_{1} \cap \operatorname {ker} f_{2}$.

Prove that the intersection of two distinct hyperplanes $H_{1}$ and $H_{2}$ of a linear space $X$ is a hyperplane in each of the hyperplanes $H_{1}$ and $H_{2}$.

Prove that if the subspaces $H_{1}$ and $H_{2}$ are hyperplanes of a real linear space $X, H_{1} \neq H_{2}$, and $h_{1} \in H_{1} \backslash H_{2}$, then $H_{1} = H_{1} \cap H_{2} \oplus R h_{1}$.

Prove that in a linear space any two of its hyperplanes are isomorphic.

The scalar product $(x, y)$ in the $n$-dimensional space $R^{n}$ is a bilinear mapping of $R^{n} \times R^{n}$ into $R$.

The vector (cross) product of three-dimensional vectors is a bilinear mapping of $R^{3} \times R^{3}$ into $R^{3}$.

The bilinear form

is a bilinear mapping of $R^{n} \times R^{n}$ into $R$.

If $z = f(x ; y), x \in X, y \in Y, z \in Z$, is a bilinear mapping of the linear space $X \times Y$ into the linear space $Z$, then for any numbers $\lambda_{1}, \lambda_{2}, \mu_{1}, \mu_{2}$ and $x_{1}, x_{2} \in X, y_{1}, y_{2} \in Y$ the equality

holds.

For any bilinear mapping $f: X \times X \rightarrow Z$ the identity

holds.

The set of all bilinear mappings of the product $X \times Y$ of linear spaces $X$ and $Y$ into a linear space $Z$ forms, under the usual definition of the linear operation on functions, a linear space.

A bilinear form $F: X \times Y \rightarrow Z$ is called symmetric if for any elements $x \in X$ and $y \in Y$ the equality $F(x ; y) = F(y ; x)$ holds. Symmetric bilinear forms $f: X^{2} \rightarrow Z$ and $g: X^{2} \rightarrow Z$, where $X^{2} = X \times X$, coincide if and only if the quadratic forms $f(x ; x)$ and $g(x ; x)$ generated by them coincide.

Prove that for any two elements $x$ and $y$ of a seminormed space the inequality

holds.

Prove that for any two elements $x$ and $y$ of a seminormed space the inequality

holds.

Prove that the seminorm of the zero element of a seminormed space equals zero.

Prove that the linear space is normed with respect to the indicated norm. $R, \left\| x\right\| = \left|x\right|$.

Prove that the linear space is normed with respect to the indicated norm. $C, \left\| z\right\| = \left|z\right|$.

Prove that the linear space is normed with respect to the indicated norm. $R^{n}$;

Prove that the linear space is normed with respect to the indicated norm. $C^{n}$ (see problem 3), $\left\| z\right\|_{2} = \sqrt{\sum_{k = 1}^{n}\left|z_{k}\right|^{2}}, z = \left(z_{1} ; \ldots ; z_{n}\right), z_{n} \in C$, $k = 1,2, \ldots$

Prove that the linear space is normed with respect to the indicated norm. $l_{p}, 1 \leqslant p < +\infty$ (see problem 12), $\left\| x\right\|_{p} = \left(\sum_{k = 1}^{\infty }\left|x_{k}\right|^{p}\right)^{1 / p}, x = \left(x_{1} ; \ldots ; x_{n} ; \ldots \right), x_{n} \in R, n = 1,2, \ldots$

Prove that the linear space is normed with respect to the indicated norm. $l_{\infty }$ (see problem 4 of § 18 and problem 7), $\left\| x\right\|_{\infty } = \sup_{k}\left|x_{k}\right|$.

Prove that the linear space is normed with respect to the indicated norm. $l_{\infty }^{(0)}$ (see problem 90 of § 18 and problem 7), $\left\| x\right\| = \max \left|x_{k}\right|$.

Prove that the linear space is normed with respect to the indicated norm. $C[a ; b]$ (see problem 8), $\left\| x\right\|_{C} = \max_{[a ; b]}\left|x(t)\right|$.

Prove that the linear space is normed with respect to the indicated norm. $C L_{p}[a ; b],-\infty < a < b < +\infty$ (see problem 7),

Prove that the linear space is normed with respect to the indicated norm. $C L_{p}(a ; b),-\infty \leqslant a < b \leqslant +\infty$ (see problem 9 of § 18),

Prove that the linear space is normed with respect to the indicated norm. $B(E)$ (see example 1 in § 18), $\left\| x\right\|_{\infty } = \sup_{E}\left|x(t)\right|$.

Prove that the linear space is normed with respect to the indicated norm. $C^{n}[a ; b]$ (see problem 93 of § 18), $\left\| x\right\|_{C^{n}} = \sum_{k = 0}^{n} \max_{[a ; b]}\left|x^{(k)}(t)\right|$.

Prove that the linear space is normed with respect to the indicated norm. $H^{\alpha }[a ; b], \alpha > 0$ (see problem 95 of § 18),

Prove that the linear space is normed with respect to the indicated norm. $H(K)$ — the set of functions analytic inside the open unit disc $K = \left\{ z \in C \mid \left|z\right| < 1\right\}$ that are continuous on its closure $\bar{K}, \left\| x\right\| = \max_{\left|z\right| = 1}\left|z(t)\right|$.

Prove the stated assertion. For the norms $\left\| x\right\|_{p}$ and $\left\| x\right\|_{\infty }$ in problems 62 and 63 the relation $\lim_{p \rightarrow \infty }\left\| x\right\|_{p} = \left\| x\right\|_{\infty }$ holds.

Prove the stated assertion. In the linear space $R^{n}$ the function $\left\| x\right\| = \sqrt{\sum_{k = 1}^{m} x_{k}^{2}}$, $0 < m < n$, is a seminorm but is not a norm on it.

Prove the stated assertion. In the linear space $C^{n}[a ; b], n > 0$ (see problem 59) the function $\left\| x\right\| = \max_{[a ; b]}\left|x^{(n)}(t)\right|$ is a seminorm but is not a norm on it.

Prove the stated assertion. The functional $\left\| x\right\|_{p} = \left(\int_{a}^{b}\left|x(t)\right|^{p} d t\right)^{1 / p}$, while being a seminorm on the space $R L_{p}[a ; b]$ (see example 2), is not a norm on it.

On which subsets of the set of functions $x(t)$ having an absolutely integrable derivative of order $n$ on the segment $[a ; b]$ is the functional $\int_{a}^{b}\left|x^{(n)}(t)\right| d t$ a norm, and on which is it a seminorm; for which $n$?

Can one take, as a norm of an element $x(t)$ in the linear space of twice continuously differentiable functions on the segment $[a ; b]$:

Can one take, as a norm of an element $x(t)$ in the linear space of continuously differentiable functions on the segment $[a ; b]$:

Prove the stated assertion. In a finite-dimensional linear space all norms are equivalent.

Prove the stated assertion. If $X$ and $Y$ are normed linear spaces, then the functionals $\max \left\{ \left\| x\right\|_{X} ;\left\| y\right\|_{Y}\right\} ,\left\| x\right\|_{X}+\left\| y\right\|_{Y}$ and $\sqrt{\left\| x\right\|_{X}^{2}+\left\| y\right\|_{Y}^{2}}$, $x \in X, y \in Y$, are equivalent norms on the space $X \times Y$.

Prove the stated assertion. The functionals $\left\| x\right\|_{C} = \max_{[a ; b]}\left|x(t)\right|$ and $\left\| x\right\|_{1} = \int_{a}^{b}\left|x(t)\right| d t$ (see problems 65 and 66) are nonequivalent norms on the linear space of functions continuous on the segment $[a ; b]$.

Prove the stated assertion. The norms $\left\| x\right\|_{C}$ and $\left\| x\right\|_{H^{1}}$ (see problems 65 and 70) are not equivalent on the set of functions satisfying the Lipschitz condition (i.e., the Hölder condition of degree 1) on the segment $[a ; b]$.

Prove the stated assertion.

Prove the stated assertion. The set $C^{n}[a ; b]$ is a non-closed subspace of the space $C[a ; b]$ (see problems 65 and 69).

Do the following sets of functions form a closed subspace of the space $C[-1 ; 1]$ (see problem 65):

For which $p$ and $q$ does the inclusion $l_{p} \subset l_{q}$ (see problem 62) hold?

Prove that if $x(t) \in R L_{p}[a ; b]$ (see example 2), $1 < p < +\infty$, then

and if $1 \leqslant p < +\infty$, then $\left\| x\right\|_{p} \leqslant (b-a)^{1 / p}\left\| x\right\|_{\infty }$ (for unbounded functions $x(t)$ it is clear that $\left\| x\right\|_{\infty } = +\infty$).

Prove that if a sequence of functions $\left(x_{n}(t)\right)$ converges uniformly to a function $x(t)$ on the segment $[a ; b]$ and $x_{n}-x \in R L_{p}[a ; b]$, then this sequence also converges to the function $x$ in the seminorm of the space $R L_{p}[a ; b]$ (see example 2).

Construct an example of a sequence of continuous nonnegative functions on the segment $[0 ; 1]$ that converges in the sense of the norm of the space $C L_{p}[0 ; 1]$ (see problem 66) but does not converge at any point of this segment.

If in a seminormed space $X$ a sequence $\left(x_{1} ; \ldots ; x_{n} ; \ldots \right)$ has a limit equal to $a \in X$, then for the same sequence to have a limit equal to $b \in X$, it is necessary and sufficient that $\left\| a-b\right\| = 0$.

If a sequence of points converges in seminorm, then it is bounded.

A function $f: X \rightarrow R$ (or $f: X \rightarrow C$), defined on a seminormed space $X$, is continuous at a point $x_{0} \in X$ if and only if for every $\varepsilon > 0$ there exists $\delta > 0$ such that for all $x \in X$ satisfying the condition $\left\| x-x_{0}\right\| < \delta$, the inequality $\left|f(x)-f\left(x_{0}\right)\right| < \varepsilon$ holds.

A seminorm is a continuous function on a seminormed space.

The operations of addition of elements and multiplication of them by a number are continuous in a seminormed space.

A set $E$ is called dense in a seminormed space $X$ if for every point $x \in X$ and every $\varepsilon > 0$ there exists a point $y \in E$ such that $\left\| y-x\right\| < \varepsilon$. The set $C_{0}^{\infty }(a ; b)$, $-\infty \leqslant a < b \leqslant +\infty$ (see problem 113 of $\S 18$), is dense in the space $R L_{p}(a ; b), 1 \leqslant p < +\infty$ (see example 2).

If $1 \leqslant p < q < +\infty$, then the set $R L_{q}[a ; b]$ is a dense subspace of the space $R L_{p}[a ; b]$ (see example 2).

Every $n$-dimensional normed space is isomorphic to $R^{n}$ (see problem 60).

Let $X$ be a seminormed space. Elements $x \in X$ and $y \in Y$ are called equivalent if $\left\| x-y\right\| = 0$. Denote by $\tilde{X}$ the set whose elements are the classes of equivalent elements of the space $X$. Let $x \in \widetilde{x} \in \tilde{X}, y \in \widetilde{y} \in \widetilde{X}$, $\lambda$ a number. Define $\widetilde{x}+\widetilde{y}$ as the element of the set $\tilde{X}$ containing $x+y$, and $\lambda \widetilde{x}$ as the element of $\tilde{X}$ containing $\lambda x$. Set $\left\| \widetilde{x}\right\|_{\tilde{X}} = \left\| x\right\|_{X}$. These definitions are well posed, i.e., they do not depend on the choice of the elements $x \in \widetilde{x}, y \in \widetilde{y}$, and the set $\tilde{X}$ is a normed linear space with the norm $\left\| \widetilde{x}\right\|_{\tilde{X}}$.

A normed space $X$ is a metric space with the metric $\rho (x ; y) = \left\| x-y\right\|_{X}$.

Give an example of a metric space in which the metric is not generated by a norm.

In the linear space of all numerical sequences, is the metric of problem 18.14 generated by some norm?

Prove the stated assertion. A set in a normed space is bounded in norm if and only if it is bounded as a set of the metric space in the sense of the metric generated by this norm.

Prove the stated assertion. The metrics generated by two norms of a linear space are equivalent to each other (see § 18, item 1) if and only if the norms generating them are equivalent.

Prove the stated assertion. A norm is a continuous function in the sense of the metric generated by this norm.

Prove the stated assertion. If in a normed space $x_{n} \rightarrow x$ and $y_{n} \rightarrow y$, then $\left\| x_{n}-y_{n}\right\| \rightarrow \left\| x-y\right\| , n \rightarrow \infty$.

Prove the stated assertion. If in a normed space $x_{n} \rightarrow x, y_{n} \rightarrow y$ and in $R \lambda_{n} \rightarrow \lambda , \mu_{n} \rightarrow \mu$, then $\lambda_{n} x_{n}+\mu_{n} y_{n} \rightarrow \lambda x+\mu y, n \rightarrow \infty$.

Prove the stated assertion. If at least one of the sets $E_{1}$ and $E_{2}$ of a normed space is open, then their algebraic sum $E_{1}+E_{2}$ is also an open set.

Prove the stated assertion. In any normed space there exist two disjoint open sets that are not contained in any disjoint closed sets.

Prove the stated assertion. Every normed space is contained in, and dense in, some Banach space (this space is called the completion of the original one).

Prove the stated assertion. All completions of a given normed space (see problem 109) are isomorphic to one another.

Prove the stated assertion. The product of Banach spaces (see problem 80) is also a Banach space.

Prove the stated assertion. A system $\left\{ x_{\alpha }\right\} , \quad \alpha \in \mathbf{U}$, of elements of a seminormed space $X$ is complete if and only if the set of finite linear combinations of its elements, i.e., its linear span, forms a dense (see problem 95) set in $X$.

Prove the stated assertion. A system $\left\{ x_{\alpha }\right\} , \alpha \in \mathbf{U}$, of elements of a normed space is complete if and only if the closure of its linear span (in the sense of the metric generated by the norm) coincides with the whole space.

Prove the stated assertion. A normed space is separable (in the sense of the metric generated by the norm) if and only if it contains a countable complete system.

Prove the stated assertion. The system of powers $1, t, t^{2}, \ldots , t^{n}, \ldots$ is complete in the space $C[a ; b]$ (see problem 8).

Prove the stated assertion.