Hilbert spaces

Prove the Cauchy-Bunyakovsky inequality. Prove that the equality $\left|(x, y)\right| = \left\| x\right\| \cdot \left\| y\right\|$ holds if and only if the vectors $x$ and $y$ are linearly dependent.

Prove that the function defined by equality (3.1) satisfies all the axioms of a norm.

Let $E$ be a Euclidean space. Prove the continuity of the inner product in each argument and jointly in both arguments.

Prove the Pythagorean theorem in a Euclidean space $E$: $x \perp y$ if and only if $\left\| x+y\right\|^{2} = \left\| x\right\|^{2}+\left\| y\right\|^{2}$. In a real Euclidean space prove the law of cosines: for any $x, y \in E$ it holds that

Let $\left\{ x_{n}\right\}_{1}^{\infty }$ and $\left\{ y_{n}\right\}_{1}^{\infty }$ be sequences of vectors of a Euclidean space, with $\left\| x_{n}\right\| \leqslant 1,\left\| y_{n}\right\| \leqslant 1$ for all $n \in \mathbb {N}$. Prove that if $\left(x_{n}, y_{n}\right) \rightarrow 1$, then $\left\| x_{n}-y_{n}\right\| \rightarrow 0$.

Let $E$ be a Euclidean space. Prove the polarization identity: for any vectors $x$ and $y$ from $E$ the equality $(x, y) = \frac{1}{4} \sum_{k = 0}^{3} i^{k}\left\| x+i^{k} y\right\|^{2}$ holds in the case of a complex space $E$, or the equality $(x, y) = \frac{1}{4}\left\| x+y\right\|^{2}-\frac{1}{4}\left\| x-y\right\|^{2}$ holds in the case of a real space $E$.

Let $\left(H_{1},(\cdot , \cdot )_{H_{1}}\right)$ and $(H_{2},(\cdot , \cdot )_{H_{2}})$ be Hilbert spaces. Denote by $\left\| \cdot \right\|_{H_{1}}$ and $\left\| \cdot \right\|_{H_{2}}$ the norms generated by the inner products. Prove that the Hilbert spaces $\left(H_{1},(\cdot , \cdot )_{H_{1}}\right)$ and $\left(H_{2},(\cdot , \cdot )_{H_{2}}\right)$ are unitarily equivalent if and only if the Banach spaces $(H_{1},\left\| \cdot \right\|_{H_{1}})$ and $(H_{2},\left\| \cdot \right\|_{H_{2}})$ are isometrically isomorphic (see Definition 2.2).

Definition 2.2: A linear mapping $J$ of a normed space $X$ into a normed space $Y$ is called an isometric embedding (isometry) if $J$ preserves the norm, i.e., $\left\| J(x)\right\|_{Y} = \left\| x\right\|_{X}$; an isometric isomorphism if $J$ is a bijective isometric embedding; an embedding if $J$ is continuous and injective; an isomorphism if $J$ is a bijective embedding and the inverse mapping $J^{-1}$ is continuous. Two normed spaces between which an isomorphism (isometric isomorphism) can be constructed are called isomorphic (respectively, isometrically isomorphic).

Prove Theorem 3.1.

Theorem 3.1 (P. Jordan, J. von Neumann, 1935): a normed space $X$ is Euclidean if and only if for any two vectors $x, y \in X$ the parallelogram equality $\left\| x-y\right\|^{2}+\left\| x+y\right\|^{2} = 2\left\| x\right\|^{2}+2\left\| y\right\|^{2}$ holds.

Prove that if in a normed space $X$ for arbitrary two vectors $x, y$ the "parallelogram inequality" $\left\| x-y\right\|^{2}+\left\| x+y\right\|^{2} \leqslant 2\left\| x\right\|^{2}+2\left\| y\right\|^{2}$ holds, then $X$ is Euclidean.

Prove that the following spaces are not Hilbert spaces:

Prove that in any Banach space $X$, for any vectors $x, y \in X$ not both equal to zero, $\frac{1}{2} \leqslant \frac{\left\| x+y\right\|^{2}+\left\| x-y\right\|^{2}}{2\left\| x\right\|^{2}+2\left\| y\right\|^{2}} \leqslant 2$ holds. Show that in both spaces $l_{\infty }(2)$ and $C[0,1]$ both extreme values are attained. ¹

Prove that the space $L_{2}(M, \Sigma , \mu )$ (see the list of spaces) is a Hilbert space.

Let $\Omega = \left\{ r_{n}\right\}_{1}^{\infty }$ be a sequence of positive numbers. Consider the set of sequences $(x_{1}, x_{2}, \ldots )$, where $x_{n} \in \mathbb {C}$, for which the series $\sum_{n = 1}^{\infty }\left(r_{n}\left|x_{n}\right|\right)^{2}$ converges. Prove that this is a linear space. Equip this set with the inner product (verify the axioms of an inner product)

and denote the resulting space $l_{2, \Omega }$. Prove that this space is $L_{2}(M, \Sigma , \mu )$, where $M = \mathbb {N}, \Sigma = 2^{\mathbb {N}}$, and $\mu (A) = \sum_{n \in A} r_{n}$. It follows that $l_{2, \Omega }$ is a Hilbert space. ¹

Prove that the Sobolev spaces $W_{2}^{n}[0,1], n \in \mathbb {N}$, are Hilbert spaces.

Prove that the Bergman space $\mathcal{A} \mathcal{L}_{2}(\left|z\right| < 1)$ is a Hilbert space.

Which of the following spaces are separable:

Consider the linear space $H$ of arbitrary functions of a real variable that are nonzero at no more than a countable number of points (the set of points is different for each function). Introduce an inner product on this space by the rule $(x, y) = \sum_{t: x(t) \cdot y(t) \neq 0} x(t) \overline{y(t)}$. Prove that $H$ is a nonseparable Hilbert space.

Prove that every Hilbert space is strictly convex, or, equivalently, 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).

Prove that every Hilbert space is uniformly convex (see Definition 2.11).

Definition 2.11: A normed space $X$ is called uniformly convex if for every $\varepsilon > 0$ there exists $\delta > 0$ such that if $\left\| x\right\| = \left\| y\right\| = 1$ and $\left\| x+y\right\| > 2-\delta$, then $\left\| x-y\right\| < \varepsilon$.

Prove that in a complex Hilbert space $H$, for any vectors $x$ and $y$, the equalities hold

(compare with problem 3.6).

Let $M$ be an arbitrary set in a Hilbert space. Prove that $M^{\perp }$ is a closed linear subspace.

Let $H_{0}$ be a proper closed subspace of a Hilbert space $H$. Prove that a vector $x$ belongs to $H_{0}^{\perp }$ if and only if for every $y \in H_{0}$ the inequality $\left\| x-y\right\| \geqslant \left\| x\right\|$ holds.

Let $H_{0}$ be a proper closed subspace of a Hilbert space $H$, and $x \in H$. Prove that $y \in H_{0}$ is the orthogonal projection of the vector $x$ if and only if $y$ is the element of best approximation for the vector $x$ in the subspace $H_{0}$.

Prove that in a Hilbert space any closed subspace is a subspace of existence (see Definition 2.17).

Definition 2.17: Let $X$ be a normed space and $Y$ a linear subspace of it. The subspace $Y$ is called a subspace of existence if for every element $x \in X$ there exists in $Y$ an element of best approximation, i.e., a vector $y_{0} \in Y$ with $\operatorname {dist}(x, Y) = \left\| x-y_{0}\right\|$.

Let $H = H_{0} \oplus_{\perp } H_{1}$, where $H_{0}$ and $H_{1}$ are closed subspaces, $x \in H$. Denote by $y$ the element of best approximation for $x$ in the subspace $H_{0}$, and by $z$ the element of best approximation for $x$ in the subspace $H_{1}$. Prove that $x = y+z$ and, correspondingly, $\operatorname {dist}\left(x, H_{0}\right) = \left\| z\right\|$, $\operatorname {dist}\left(x, H_{1}\right) = \left\| y\right\| , \operatorname {dist}^{2}\left(x, H_{0}\right) + \operatorname {dist}^{2}\left(x, H_{1}\right) = \left\| x\right\|^{2}$.

Let $H_{0}$ be a closed subspace of a Hilbert space $H$. Prove that $H = H_{0} \oplus_{1} H_{0}^{\perp }$. Is this true in a pre-Hilbert space $H$?

Let $H$ be a Hilbert space, $H_{0}$ a proper closed subspace, $H / H_{0}$ the quotient space. Prove that the mapping $J: H_{0}^{\perp } \rightarrow H / H_{0}, J x = [x]$ is an isometric isomorphism (here $[x]$ is the quotient class from $H / H_{0}$ containing the vector $x$).

Prove that a linear subspace $M$ in a Hilbert space $H$ is everywhere dense in $H$ if and only if $M^{\perp } = \left\{ 0\right\}$.

Prove that in a Hilbert space, for any set $M$, the equality $\left(M^{\perp }\right)^{\perp } = \overline{\operatorname {Lin}(M)}$ (the closure of the linear span of the set $M$) holds.

Let $H$ be a Hilbert space, and let $M$ and $N$ be two mutually orthogonal linear subspaces of it. Prove that their orthogonal direct sum is a topological direct sum. Prove that if they are both closed, then $M \oplus_{\perp } N$ is a closed subspace in $H$.

In a Hilbert space $H$ give examples of closed subspaces $M$ and $N$ such that $H = M \oplus N$, but this topological direct sum is not an orthogonal direct sum.

Let $H$ be a Hilbert space, $\left\{ x_{n}\right\}_{1}^{\infty }$ a sequence of vectors from $H$, and $\left\{ \lambda_{n}\right\}_{1}^{\infty }$ a sequence of complex numbers. Prove that the set $M = \left\{ x \in H:\left(x, x_{k}\right) = \lambda_{k}, k \in \mathbb {N}\right\}$ is either empty or is a closed affine subspace in $H$. ¹

Prove that any closed convex set in a Hilbert space is a Chebyshev set. ¹

Give an example of a sequence of nonempty nested bounded closed sets in the space $l_{2}$ having empty intersection.

Prove that the angle $\widehat{x, M}$ between a subspace $M$ and a vector $x$ in a real Hilbert space $H$ takes values in the interval $\left[0, \frac{\pi }{2}\right]$.

In the space $L_{2}[-1,1]$, for the following sets find the convex hull, the closure, the linear span, the closure of the linear span, and the orthogonal complement: ¹

In the space $W_{2}^{1}[-1,1]$, for the following sets find the convex hull, the closure, the linear span, the closure of the linear span, and the orthogonal complement:

Let $H_{n}, n \in \mathbb {N} \cup \left\{ \infty \right\}$, be the set of vectors $x = \left(x_{1}, x_{2}, \ldots \right)$ of the space $l_{2}$ for which $\sum_{k = 1}^{n} x_{n} = 0$. Find $H_{n}^{\perp }$ and prove that all $H_{n}, n \neq \infty$, are closed subspaces, while $H_{\infty }$ is a non-closed subspace, everywhere dense in $l_{2}$.

In the real spaces $L_{2}[0,1]$ and $W_{2}^{n}[0,1], n \in \mathbb {N}$, find the angle between the vectors $x(t) = 1$ and $y(t) = t$.

In the space $L_{2}[0,1]$ consider the continuous curve $\gamma :[0,1] \rightarrow L_{2}[0,1], \gamma (\tau ) = \chi_{\tau }$, where $\chi_{a}$ is the characteristic function of the segment $[0, a]$. Let $x = \gamma \left(\tau_{2}\right)-\gamma \left(\tau_{1}\right), y = \gamma \left(\tau_{3}\right)-\gamma \left(\tau_{2}\right)$, $0 \leqslant \tau_{1} < \tau_{2} < \tau_{3} \leqslant 1$, — two chords of the curve $\gamma$ with a common endpoint. Prove that $x$ and $y$ are orthogonal. Does such a curve exist in a finite-dimensional Euclidean space?

In the space $L_{2}[0,1]$ find the distance from the vector $x(t) = t^{n}, n \in \mathbb {N}$, to the subspace $H_{0} = \left\{ x \in L_{2}[0,1]: \int_{0}^{1} x(t) d t = 0\right\}$.

Let $H_{0}$ be a finite-dimensional subspace of a Hilbert space $H$, and let $\left\{ h_{k}\right\}_{1}^{n}$ be a linear basis (not necessarily orthogonal) in $H_{0}$. Prove that for any $x \in H$ the element of best approximation in $H_{0}$ has the form

and the distance from $x$ to $H_{0}$ is given by the equality

In particular, if $\left\{ h_{k}\right\}_{1}^{n}$ is an orthonormal basis in $H_{0}$, then the element of best approximation is $y = \sum_{k = 1}^{n}\left(x, h_{k}\right) h_{k}$, and

In the space $l_{2}$ find $\operatorname {dist}\left(e_{1}, H_{n}\right)$, where $e_{1} = (1,0,0, \ldots )$, $H_{n} = \left\{ x \in l_{2}: \sum_{k = 1}^{n} x_{k} = 0\right\} , n \in \mathbb {N}$.

In the space $L_{2}[0,1]$ find the distance from the vector $x(t) = t^{2}$ to the subspace $\mathcal{P}_{1}$ of all linear functions. Find the element of best approximation for $x$ in $\mathcal{P}_{1}$ (compare with problem 2.73).

Prove that a system of finitely many vectors $\left\{ x_{k}\right\}_{k = 1}^{n}$ of a Hilbert space is linearly independent if and only if its Gram determinant is nonzero.

Apply the Gram-Schmidt orthogonalization process to the system $\left\{ 1, t, t^{2}\right\}$ in the spaces

Apply the Gram-Schmidt orthogonalization process to the system $\left\{ e^{-t^{2} / 2}, t e^{-t^{2} / 2}, t^{2} e^{-t^{2} / 2}, t^{3} e^{-t^{2} / 2}\right\}$ in the space $L_{2}(\mathbb {R})$.

Let $H$ be a separable infinite-dimensional Hilbert space, and let $\left\{ e_{n}\right\}$ be an arbitrary orthonormal system in it. Prove that this system can be completed to an orthonormal basis $\left\{ e_{n}\right\}_{1}^{\infty }$. Deduce from this that in any separable Hilbert space there exists an orthonormal basis.

Prove that the following systems are orthonormal bases in the space $L_{2}[-\pi , \pi ]$:

Prove that the following systems are orthonormal bases in the space $L_{2}[0, \pi ]$:

Prove that the system of Legendre polynomials

is an orthonormal basis in the space $L_{2}[-1,1]$.

Prove that the system of Chebyshev polynomials of the first kind (see problem 2.73)

is an orthonormal basis in the space $L_{2}\left([-1,1], \frac{d t}{\sqrt{1-t^{2}}}\right)$.

Prove that the system of Jacobi polynomials

(here $\alpha > -1, \beta > -1, n = 0,1, \ldots$) is an orthonormal basis in the space $L_{2}\left([-1,1],(1-t)^{\alpha }(1+t)^{\beta } d t\right)$.

Prove that the system of Laguerre polynomials

is an orthonormal basis in the space $L_{2}\left((0, \infty ), e^{-t} d t\right)$.

Prove that the system of Hermite polynomials

is an orthonormal basis in the space $L_{2}\left(\mathbb {R}, e^{-t^{2}} d t\right)$.

Find the orthonormal system obtained by applying the Gram-Schmidt orthogonalization process to the system $\left\{ e^{i n t}\right\}_{n \in \mathbb {Z}}$ in the space $W_{2}^{1}[-\pi , \pi ]$. Is the resulting system an orthonormal basis in this space?

Is the orthonormal system obtained in the previous problem a basis in the space $H: = \left\{ f \in W_{2}^{1}[-\pi , \pi ]: f(-\pi ) = f(\pi )\right\}$?

Find the orthonormal system $\left\{ \varphi_{n}\right\}$ obtained by applying the Gram-Schmidt orthogonalization process to the system $\left\{ \sin n t\right\}_{n = 1}^{\infty }$ in the space $W_{2}^{1}[0, \pi ]$. Is the resulting system an orthonormal basis:

Prove that the set of sequences of Fourier coefficients of all functions from the space $\dot{W}_{2}^{1}[0,1]$ with respect to the system $\left\{ \varphi_{n}\right\}$ from the previous problem coincides with the linear space $l_{2, \mathbb {N}}$ (see the definition in problem 3.13).

Find the orthonormal system obtained by applying the Gram-Schmidt orthogonalization process to the system $\left\{ \cos n t\right\}_{n = 0}^{\infty }$ in the space $W_{2}^{1}[0, \pi ]$. Is the resulting system an orthonormal basis in this space?

Find the orthonormal system obtained by applying the Gram-Schmidt orthogonalization process to the system $\left\{ \cos n t, \sin (n+1 / 2) t\right\}_{n = 0}^{\infty }$ in the space $W_{2}^{1}[-\pi , \pi ]$. Is the resulting system an orthonormal basis in this space?

Find the orthonormal system obtained by applying the Gram-Schmidt orthogonalization process to the system $\left\{ \zeta^{n}\right\}_{n = 0}^{\infty }$ in the space $\mathcal{A} \mathcal{L}_{2}(\left|\zeta \right| < 1)$. Is the resulting system an orthonormal basis in this space?

Prove that the Rademacher system

is orthonormal, but not complete in $L_{2}[0,1]$.

Prove that the Walsh system, consisting of all possible finite products of functions from the Rademacher system, is an orthonormal basis in $L_{2}[0,1]$.

Prove that the Haar system

where $n$ is the integer part of $\log_{2} m$, and $k = m-2^{n}$, is an orthonormal basis in $L_{2}[0,1]$.

Let $\left\{ x_{n}\right\}_{1}^{\infty }$ and $\left\{ y_{n}\right\}_{1}^{\infty }$ be orthonormal bases in the spaces $L_{2}\left(M_{1}, \Sigma_{1}, \mu_{1}\right)$ and $L_{2}\left(M_{2}, \Sigma_{2}, \mu_{2}\right)$ respectively. Prove that $\left\{ x_{n} \cdot y_{m}\right\}_{m, n = 1}^{\infty }$ is an orthonormal basis in the space $L_{2}(M, \Sigma , \mu )$, where $(M, \Sigma , \mu ) = \left(M_{1}, \Sigma_{1}, \mu_{1}\right) \times \left(M_{2}, \Sigma_{2}, \mu_{2}\right).{ }$¹

Exhibit some orthonormal basis in the space $L_{2}(\mathbb {R})$.

(N. K. Bari, 1951). Let $\left\{ e_{n}\right\}_{1}^{\infty }$ be an orthonormal basis in a Hilbert space $H$, and let the orthonormal system $\left\{ f_{n}\right\}_{1}^{\infty }$ be such that $\sum_{n = 1}^{\infty }\left\| e_{n}-f_{n}\right\|^{2} < \infty$. Prove that this system is also an orthonormal basis in $H$.

Let a set $M$ in a Hilbert space $H$ have the following property: for every $x \in H$ the set $\left\{ (x, y): y \in M\right\} \subset \mathbb {C}$ is bounded. Prove that then the set $M$ itself is bounded in the space $H$.

Using Parseval's equality, compute the sums of the series:

Let $\left\{ x_{n}\right\}_{1}^{\infty }$ be an orthogonal system of vectors of a Hilbert space $H$. Prove that the following conditions are equivalent:

1. the series $\sum_{n = 1}^{\infty } x_{n}$ converges;

2. for every $y \in H$ the series $\sum_{n = 1}^{\infty }\left(x_{n}, y\right)$ converges; ¹

3. the numerical series $\sum_{n = 1}^{\infty }\left\| x_{n}\right\|^{2}$ converges.

Definition 7.1: A sequence $\left\{ x_{n}\right\}_{1}^{\infty }$ in a normed space $X$ converges weakly to a vector $x \in X$ if $f\left(x_{n}\right) \rightarrow f(x)$ for every $f \in X^{*}$. Notation: $x_{n} \rightharpoonup x$.