Hilbert Spaces

Prove the stated assertion. For every element $x$ of a space with an almost-scalar (semi-inner) product, the equality $(x, 0) = 0$ holds.

Prove the stated assertion. An almost-scalar product in a real linear space is a bilinear mapping, but in a complex space it is not.

Prove the stated assertion. For an almost-scalar product in a real (complex) linear space the inequality

holds (the Cauchy--Bunyakovsky inequality).

Prove the stated assertion. For an almost-scalar product the inequality

holds.

Prove the stated assertion. If $(x, y)$ is an almost-scalar (scalar) product in a linear space, then the functional $\left\| x\right\| = \sqrt{(x, x)}$ is a seminorm (norm) on this space.

Prove the stated assertion. In the set of real numbers $R$ the ordinary operation of multiplication is a scalar product, and in the set of complex numbers $C$ the scalar product of numbers $z_{1}$ and $z_{2}$ is the product $z_{1} \overline{z_{2}}$.

Prove the stated assertion. In the real $n$-dimensional vector space $R^{n}$ the functional

is a scalar product, and the norm corresponding to it coincides with the length of the vector.

Prove the stated assertion. In the complex $n$-dimensional vector space $C^{n}$ (see problem 3) the function

is a scalar product.

Prove the stated assertion. In the real $n$-dimensional vector space $R^{n}$ the function

is an almost-scalar product and is not a scalar product.

Prove the stated assertion. In the linear space $R L_{2}(a ; b),-\infty \leqslant a < b \leqslant +\infty$, consisting of real functions whose square is integrable (generally, in the improper sense) on the interval $(a ; b)$, the functional

is an almost-scalar product and is not a scalar product.

Prove the stated assertion. In the linear space of functions $x:[a ; b] \rightarrow R$ continuous on the segment $[a ; b]$, the functional

is a scalar product. The resulting space with the scalar product is denoted $C L_{2}[a ; b]$.

Prove the stated assertion. In the linear space consisting of complex-valued functions whose squared modulus is integrable on a finite or infinite interval, the functional

is an almost-scalar product and is not a scalar product.

Prove the stated assertion. In the linear space of complex-valued functions continuous on the segment $[a ; b]$, the functional (9) is a scalar product.

Prove the stated assertion. In the space $C L_{2}(a ; b)$ of real functions continuous on a finite or infinite interval $(a ; b),-\infty \leqslant a < b \leqslant +\infty$, whose square is integrable on this interval, the functional

is a scalar product.

Prove the stated assertion. In the linear space $l_{2}$ of real numerical sequences (see example 2 in § 18) the functional

is a scalar product. Give an example of an almost-scalar product in this linear space.

Prove the stated assertion. In the linear space of sequences $\left(x_{1} ; \ldots ; x_{n} ; \ldots \right)$ of complex numbers (see problem 13 of § 19) for which $\sum_{n = 1}^{\infty }\left|x_{n}\right|^{2} < +\infty$, the functional

is a scalar product.

Prove the stated assertion. In the linear space $l_{2}$ (see problem 15) the functional

for $m > 1$ is an almost-scalar product and is not a scalar product.

Prove the stated assertion. Let $X$ be a linear space with an almost-scalar product. Elements $x \in X, y \in X$ are called equivalent if $\left\| x-y\right\|^{2} \equiv (x-y, x-y) = 0$. Denote by $X$ the set whose elements are the classes of equivalent elements of the space $X$. Let $x \in \widetilde{x} \in X, y \in \widetilde{y} \in \widetilde{X}, \lambda$ and $\mu$ be numbers. Define $\lambda \widetilde{x}+\mu \widetilde{y}$ as the element of the set $\widetilde{X}$ containing $\lambda x+\mu y$, and set $(\widetilde{x}, \widetilde{y}) = (x, y)$. Then 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 $X$ is a linear space, with the function $(\widetilde{x}, \widetilde{y})$ a scalar product on it.

Find the angles of the triangle with vertices at the points $x_{1}(t) \equiv 0$, $x_{2}(t) \equiv 1, x_{3}(t) \equiv t$ in the space $C L_{2}[-1 ; 1]$ (see problem 11 of § 20).

In a linear space with an almost-scalar product, for any two elements $x$ and $y$ of the space the equality

holds (the parallelogram equality).

In a linear space with an almost-scalar product, for any three elements $x, y$ and $z$ of the space the equality

holds (the Apollonius equality).

In a real Banach space a scalar product $(x ; y)$ for which $\left\| x\right\|^{2} = (x, x)$ can be introduced if and only if for any points $x, y$ of this space the equality

holds.

In the normed space $C[a ; b]$ one cannot introduce a scalar product consistent with the norm of this space in the sense of problem 22.

The operations of addition of elements and multiplication of them by a number are continuous in a space with an almost-scalar product.

An almost-scalar product in a linear space $X$ is a continuous function on $X$.

If in a linear space $X$ with an almost-scalar product a convergent series $\sum_{n = 1}^{\infty } x_{n} = x, x_{n} \in X$, is given, then for every element $a \in X$ the numerical series obtained from it by term-by-term multiplication by $a$ also converges, and $\sum_{n = 1}^{\infty }\left(x_{n}, a\right) = (x, a)$.

If in the space $R L_{2}(a ; b)$ (see problem 10) the series $\sum_{n = 1}^{\infty } x_{n}(t)$ converges and its sum equals $x(t)$, i.e., $\sum_{n = 1}^{\infty } x_{n}(t) = x(t)$, then for every function $\varphi (t) \in R L_{2}[a ; b]$ the equality

holds; in particular, for a finite interval $(a, b)$ the equality

holds, i.e., the given series can be integrated term by term.

All $n$-dimensional linear spaces with a scalar product are isomorphic to one another.

Every $n$-dimensional linear space with a scalar product is complete in the sense of the metric generated by the scalar product.

Every linear space with a scalar product that is isomorphic to a Hilbert space is itself a Hilbert space.

Every linear space (real or complex) with a scalar product is contained in, and dense in, some Hilbert space, called its completion.

All completions of a linear space with a scalar product are isomorphic to one another.

The set $\left\{ x = \left(x_{1} ; \ldots ; x_{n} ; \ldots \right) \in l_{2} \mid \sum_{n = 1}^{\infty } x_{n} = 0\right\}$ is a linear space dense in the space $l_{2}$ (see problem 15).

The space $C L_{2}[a ; b]$ (see problem 11) is not a Hilbert space.

The space $C L_{2}[a ; b]$ (see problem 11) is dense in some Hilbert space (this space is denoted $L_{2}[a ; b]$).

In the linear space $C L_{2}^{1}[a ; b]$ of continuously differentiable functions on the segment $[a ; b]$, will the functional

Find the angle between the functions $x(t) = \sin t$ and $y(t) = t$ in the space:

The inclusion (see § 19, item 2) $C[a ; b] \subset L_{2}[a ; b]$ (see problem 35) holds.

The space $L_{2}[a ; b]$ (see problem 35) is separable.

If $\lim_{n \rightarrow \infty } \int_{a}^{b}\left[x(t)-x_{n}(t)\right]^{2} d t = 0$, then the sequence of functions $\left\{ x_{n}(t)\right\}$ is called convergent in the mean-square sense on the segment $[a ; b]$ to the function $x(t)$.

The function $x(t) = 1 / \sqrt{t}$ is not the limit, in the mean-square sense on the segment $[0 ; 1]$, of a sequence of continuous functions.

The space $l_{2}$ (see problem 15) is Hilbert.

If a system $\left\{ x_{\alpha }\right\} , \alpha \in \mathbf{U}$, of elements of a linear space with an almost-scalar product is orthogonal and $\left\| x_{\alpha }\right\| \neq 0$ for all $\alpha \in \mathbf{U}$, then it is linearly independent.

If elements $x_{1}, x_{2}, \ldots$ and $y_{1}, y_{2}, \ldots$ of a linear space with an almost-scalar product are such that

(such systems are called biorthogonal), then each of these systems is linearly independent.

For a system of elements $x_{1}, x_{2}, \ldots , x_{n}$ of a linear space with a scalar product to be linearly dependent, it is necessary and sufficient that its Gram determinant

be equal to zero.

If $\left\{ e_{\alpha }\right\} , \alpha \in \mathbf{U},$ is an orthonormal system in a linear space with a scalar product, then for any $\alpha \in \mathbf{U}$ and $\beta \in \mathbf{U}, \alpha \neq \beta$, the equality $\left\| e_{\alpha }-e_{\beta }\right\| = \sqrt{2}$ holds.

If a linear space with a scalar product is separable, then every orthonormal system in it is at most countable.

The trigonometric system of functions

is orthogonal in the space $L_{2}[-\pi ; \pi ]$ (see problem 35), and the corresponding orthonormal system has the form

The Legendre polynomials $P_{n}(t), n = 0,1,2, \ldots$ (see problem 28 of $\S 19)$ form an orthogonal system in the space $L_{2}[-1 ; 1]$ (see problem 35).

For any segment $[a ; b]$, the Legendre polynomials $P_{n}(t), n = 0,1,2, \ldots$ (see problem 28 of § 19) form a complete system in the space $L_{2}[a ; b]$ (see problem 35).

The system of functions $\left\{ e^{i n t}\right\} , n = 0, \pm 1, \pm 2, \ldots$, forms a complete orthogonal system in the space of complex-valued functions continuous on the segment $[-\pi ; \pi ]$ with the scalar product (see problem 13) $(x, y) = \int_{-\pi }^{\pi } x(t) \overline{y(t)} d t$.

The sequence of functions $\sin (2 n-1) t / 2, n = 1,2, \ldots$, forms an orthogonal system in the space $L_{2}[0 ; \pi ]$ (see problem 35).

The functions $\sin t, \sin 3 t, \sin 5 t, \sin 7 t, \sin 9 t$ are linearly independent.

If $x_{1}, x_{2}, \ldots , x_{n}$ is an orthogonal system in a linear space with a scalar product and $x = x_{1}+x_{2}+\ldots +x_{n}$, then $\left\| x\right\|^{2} = \sum_{k = 1}^{n}\left\| x_{k}\right\|^{2}$.

If $x_{1}, x_{2}, \ldots , x_{n}, \ldots$ is an orthogonal system in a Hilbert space, then the series $\sum_{n = 1}^{\infty } x_{n}$ converges in this space if and only if the numerical series $\sum_{n = 1}^{\infty }\left\| x_{n}\right\|^{2}$ converges.

If $x_{n}, n = 1,2, \ldots$, is a linearly independent system of elements of a linear space $X$ with a scalar product, then there exists an orthogonal system of elements $y_{n}, y_{n} \neq 0, n = 1,2, \ldots$, of this space such that

(here $\alpha_{n k}$ are real numbers if the linear space under consideration is real, and complex if it is complex).

Constructing the system $\left(y_{1} ; \ldots ; y_{n} ; \ldots \right)$ from the system $\left(x_{1} ; \ldots ; x_{n} ; \ldots \right)$ is called orthogonalizing the latter.

Carry out the orthogonalization of the system of functions $x_{0}(t) = 1$, $x_{1}(t) = t, x_{2}(t) = t^{2}, x_{3}(t) = t^{3}$ in the space:

Prove the stated assertion. If, under the conditions of problem $55\left(z_{1} ; \ldots ; z_{n} ; \ldots \right)$ is an orthogonal system and $z_{n} = \beta_{n 1} x_{1}+\ldots +\beta_{n n} x_{n}$, then $y_{n}$ and $z_{n}$ differ from each other by a scalar factor: $z_{n} = \lambda_{n} y_{n}, n = 1,2, \ldots$

Prove the stated assertion. Orthogonalizing (see problem 55) the system of powers $1, t, t^{2}, \ldots , t^{2}, \ldots$ in the space $L_{2}[-1 ; 1]$ (see problem 35) yields polynomials that can differ from the Legendre polynomials (see problem 28 of § 19) only by numerical factors.

Prove the stated assertion. Let $e_{k}, k = 1,2, \ldots , n$, be a finite orthogonal system of elements of a linear space $X$ with a scalar product, and let $\mathscr {L}\left(e_{1} ; \ldots ; e_{n}\right)$ be the linear span of these elements. Then the element of best approximation of an element $x \in X$ in the subspace $\mathscr {L}\left(e_{1} ; \ldots ; e_{n}\right)$ has the form $x_{0} = \sum_{k = 1}^{n} a_{k} e_{k}$, where $a_{k}$ are the Fourier

coefficients of the element $x$ with respect to the system $\left\{ e_{k}\right\} , k = 1,2, \ldots , n$. Moreover,

Prove the stated assertion. The element $x_{0} = \sum_{j = 1}^{n} \alpha_{j} e_{j}$ is the element of best approximation of an element $x \in X$ in the subspace $\mathscr {L}\left(e_{1} ; \ldots ; e_{n}\right) \in X$ if and only if the element $x-x_{0}$ is orthogonal to all elements of $\mathscr {L}\left(e_{1} ; \ldots ; e_{n}\right)$, which is written as $x-x_{0} \perp \mathscr {L}\left(e_{1} ; \ldots ; e_{n}\right)$. (For the generalization of this problem to the case of an infinite set $\left\{ e_{n}\right\}$ see problem 91.)

Prove the stated assertion. If $e_{n} \subset X, n = 1,2, \ldots , x \in X$ and $E_{n}(x)$ is the best approximation of the element $x$ by the space $\mathscr {L}\left(e_{1} ; e_{2} ; \ldots ; e_{n}\right)$, then $E_{n+1}(x) \leqslant E_{n}(x)$.

Prove the stated assertion. If $e_{n} \in X, e_{n} \neq 0, n = 1,2, \ldots$, is an orthogonal system in a linear space with a scalar product $X$, then the partial sums $s_{n} = \sum_{k = 1}^{n} a_{k} e_{k}$ of the Fourier series of the element $x$ are elements of its best approximation in the space $\mathscr {L}\left(e_{1} ; e_{2} ; \ldots ; e_{n}\right)$.

Prove the stated assertion. If $s_{n}$ are the partial sums of the Fourier series of an element $x$ of a linear space with a scalar product $X$ with respect to an orthogonal system $e_{n} \in X, e_{n} \neq 0, n = 1,2, \ldots$, then the numerical sequence $\left\{ \left\| x-s_{n}\right\| \right\}$ is decreasing.

Prove the stated assertion. For the Fourier coefficients $a_{n}$ of an element $x$ of a linear space with a scalar product $X$ with respect to an orthogonal system $e_{n} \in X, e_{n} \neq 0, n = 1,2, \ldots$, the inequality

holds (Bessel's inequality).

Prove the stated assertion. If there exists a constant $c > 0$ such that for all elements of an orthogonal system $e_{n}, n = 1,2, \ldots$, of a linear space with a scalar product $X$ the inequalities $\left\| e_{n}\right\| \geqslant c$ hold (in particular, if this system is orthonormal), then the Fourier coefficients of every element $x \in X$ with respect to this system tend to zero as $n \rightarrow \infty$.

Prove the stated assertion. If the space $X$ is Hilbert, then the Fourier series $\sum_{n = 1}^{\infty } a_{n} e_{n}$ of every element $x \in X$ with respect to any orthogonal system $e_{n} \in X$, $e_{n} \neq 0, n = 1,2, \ldots$, converges in the space $X$. If $x_{0} = \sum_{n = 1}^{\infty } a_{n} e_{n}$ is the sum of this series, then the element $x-x_{0}$ is orthogonal to all elements of the system $\left(e_{1} ; \ldots ; e_{n} ; \ldots \right)$.

Prove the stated assertion. The Fourier series of an element $x$ of a linear space with a scalar product $X$ with respect to an orthogonal system $e_{n} \in X, e_{n} \neq 0, n = 1,2, \ldots$, converges to this element if and only if the equality

holds (Parseval's equality), where $a_{n}$ are the Fourier coefficients of the element $x$ with respect to the system $\left(e_{1} ; \ldots ; e_{n} ; \ldots \right)$.

Prove the stated assertion. The Fourier series with respect to an orthogonal system of every element of a linear space with a scalar product converges to that element itself if and only if the given orthogonal system is complete.

Prove the stated assertion. For an orthogonal system to be complete in a linear space with a scalar product, it is necessary and sufficient that for every element of the space Parseval's equality (10) holds with respect to this system.

Prove the stated assertion. If an orthogonal system is complete in a linear space with a scalar product, then an element of the space all of whose Fourier coefficients with respect to this system are zero is itself zero.

Prove the stated assertion. From the equality of all the Fourier coefficients of two elements of a linear space with a scalar product with respect to a complete orthogonal system, it follows that these elements are equal.

Prove the stated assertion. If in a linear space with a scalar product $X$, an element $x \in X$ admits a representation of the form $x = \sum_{n = 1}^{\infty } \lambda_{n} e_{n}$, where $c_{n} \in X, e_{n} \neq 0, n = 1,2, \ldots$, is an orthogonal system, then this representation is unique and the coefficients $\lambda_{n}$ are the Fourier coefficients of the element $x$ with respect to the system $\left(e_{1} ; \ldots ; e_{n} ; \ldots \right)$.

Prove the stated assertion. Every complete orthogonal system $e_{n} \in X, e_{n} \neq 0, n = 1,2, \ldots$, is a basis in a linear space with a scalar product.

Prove the stated assertion. In a Hilbert space $X$ an orthogonal system $e_{n} \in X, e_{n} \neq 0, n = 1,2, \ldots$, is complete if and only if it is closed.

Give an example of a closed system in some linear space with a scalar product that is not complete. Do there exist complete systems that are not closed?

Prove the stated assertion. The elements $e_{n} = \left(x_{1}^{(n)} ; \ldots ; x_{m}^{(n)} ; \ldots \right), x_{n}^{(n)} = 1, x_{m}^{(n)} = 0$ for $m \neq n, n = 1,2, \ldots$, form an orthonormal basis in the space $l_{2}$ (see problem 15).

Prove the stated assertion. The Legendre polynomials (see problem 28 of § 19) form an orthogonal basis in the space $L_{2}[-1 ; 1]$ (see problem 35).

Prove the stated assertion. The trigonometric system $1, \cos t, \sin t, \ldots , \cos n t, \sin n t, \ldots$ forms an orthogonal basis in the space $L_{2}[-\pi ; \pi ]$ (see problem 35).

Prove the stated assertion. The functions $\frac{1}{\sqrt{b-a}} \exp \left\{ \frac{2 \pi i n(t-a)}{b-a}\right\} , \quad n = 0,1,2, \ldots$, form an orthonormal basis in the space that is the completion (see problem 31) of the space of complex-valued functions continuous on the segment $[a, b]$ with the scalar product (9).

Prove the stated assertion. The functions $\sqrt{2 / \pi } \sin n t, n = 1,2, \ldots$, form an orthonormal basis in the space $L_{2}[0 ; \pi ]$, while in the space $L_{2}[-\pi ; \pi ]$ (see problem 35) they are an orthogonal system but not a basis.

Prove the stated assertion. The functions $1, \sin \frac{2 \pi n(t-a)}{b-a}, \cos \frac{2 \pi n(t-a)}{b-a}, \quad n = 1,2, \ldots$, are orthogonal in the space $C L_{2}^{1}[a ; b]$ (see problem 36).

Prove the stated assertion. Every function $x \in R L_{2}[-\pi ; \pi ]$ (see problem 10) expands into a Fourier series with respect to the trigonometric system of functions, convergent in the mean-square sense (see problem 40):

moreover Parseval's equality

holds.

Prove the stated assertion. If for a function $x \in R L_{2}[-\pi ; \pi ]$ (see problem 10) all its Fourier coefficients with respect to the trigonometric system are zero, then it is equivalent to zero (see problem 88 of § 19).

Prove the stated assertion. In every separable linear space with a scalar product there exists an orthonormal basis.

Prove the stated assertion. The space $l_{2}$ (see problem 15) is separable.

Prove the stated assertion. All separable infinite-dimensional Hilbert spaces are isomorphic to one another.

Prove the stated assertion. If a subspace $Z$ of a space $X$ is the orthogonal complement of a subspace $Y$ of the same space, then $Y$ is also the orthogonal complement of the subspace $Z$.

Prove the stated assertion. If the sets $Y$ and $Z$ are closed subspaces of a space $X$, then their sum (see § 19, item 1) $Y+Z$ is also a closed subspace of the space $X$.

Prove the stated assertion. For a subspace $Y$ of a space $X$ to be dense in this space, it is necessary and sufficient that the condition $x \perp Y$ (i.e., $(x, y) = 0$ for all $y \in Y)$ imply that $x = 0$.

Prove the stated assertion. If $Y$ is a closed subspace of a Hilbert space $X$ and $x_{0} \in X$, then there exists a unique element $y_{0} \in Y$ such that

(the element $y_{0}$ is called the orthogonal projection of the element $x_{0}$ onto the space $Y$).

Prove the stated assertion. For an element $y_{0}$ to be the orthogonal projection of an element $x_{0}$ of a Hilbert space $X$ onto its closed subspace $Y$ (see problem 90), it is necessary and sufficient that for all $y \in Y$ the condition $\left(x_{0}-y_{0}, y\right) = 0$ hold.

Prove the stated assertion. If $Y^{\perp }$ is the orthogonal complement of a closed subspace $Y$ of a Hilbert space $X$, then

moreover, if $x_{0} = y_{0}+z_{0} \in X, y_{0} \in Y, z_{0} \in Y^{\perp }$, then

Prove the stated assertion. For an element $x$ of a space $X$ to be orthogonal to a subspace $Y \subset X$, it is necessary and sufficient that for every element $y \in Y$ the inequality $\left\| x\right\| \leqslant \left\| x-y\right\|$ hold.

Prove the stated assertion. For any subset $E$ of a space $X$, the set $E^{\perp }$ is a closed subspace of $X$.

Prove the stated assertion. If $E$ is a subset of a space $X$ with a scalar product, then the inclusion $E \subset \left(E^{\perp }\right)^{\perp }$ holds. Can this inclusion be strict?

Prove the stated assertion. For a subset $E$ of a space $X$, the equality $\left(E^{\perp }\right)^{\perp } = E$ holds if and only if the subset $E$ is a closed subspace of the space $X$.

Prove the stated assertion. If $E_{1} \subset E_{2} \subset X$, then $E_{1}^{\perp } \supset E_{2}^{\perp }$.

If $X$ is a Hilbert space and $X = Y \oplus Z$, does it follow that $Z = Y^{\perp }$? And in the case of a finite-dimensional space?

Prove the stated assertion. If $Y = \left\{ x \in C L_{2}[-1 ; 1] \mid x(t) = 0 \text{ for all } t \in [0 ; 1]\right\}$ (see problem 14), then $Y$ is a subspace of the space $C L_{2}[-1 ; 1]$. Describe the space $Y^{\perp }$. Does the decomposition $C L_{2}[-1 ; 1] = Y \oplus Y^{\perp }$ hold?

Prove the stated assertion. The set $Y = \left\{ x(t) \in C L_{2}^{1}[a ; b] \mid \int_{a}^{b} x(t) d t = 0\right\}$ is a subspace of the space $C L_{2}^{1}[a ; b]$ (see problem 36). Find $Y^{\perp }$.

Prove the stated assertion. If

then $\overline{Y+Z} = l_{2}$, but $Y+Z \neq l_{2}$, and hence $Y+Z$ is not a closed subspace of the space $l_{2}$.

Prove the stated assertion. For every bounded linear functional $f$ on a real (complex) Hilbert space $X$ there exists a unique element $a \in X$ such that for all $x \in X$ the equality $f(x) = (x, a)$ holds, and moreover $\left\| f\right\| = \left\| a\right\|$.

Prove the stated assertion. If $A$ is a bounded linear operator on a linear space $X$ with a scalar product, then

Prove the stated assertion. If $A$ is a bounded linear operator on a linear space $X$ with a scalar product, then the function $f(x ; y) = (A x, y)$ is a bilinear functional and $\left\| f\right\| = \left\| A\right\|$.

Prove the stated assertion. For every bounded bilinear functional $f$ on a Hilbert space $X$ there exists a unique bounded linear operator $A$ such that $f(x ; y) = (A x, y)$ for all $x \in X$, $y \in X$.