Bases in Hilbert spaces

Prove that every orthonormal system of vectors of a Hilbert space is uniformly minimal.

Prove Theorem 10.2.

Theorem 10.2: let $\left\{ x_{k}\right\}_{1}^{\infty }$ be a system of vectors of a Hilbert space $H$. Then the following conditions are equivalent:

1. the system $\left\{ x_{k}\right\}_{1}^{\infty }$ is a Riesz basis;

2. there exists a new inner product $\langle x, y\rangle$, equivalent to the standard one (i.e., the norms $\sqrt{\langle x, x\rangle }$ and $\sqrt{(x, x)}$ are equivalent), such that the system $\left\{ x_{k}\right\}_{1}^{\infty }$ is an orthonormal basis of the space $H$ with respect to the new inner product;

3. the system $\left\{ x_{k}\right\}_{1}^{\infty }$ is complete and possesses the Bessel property: there exist constants $\gamma_{1}, \gamma_{2} > 0$ such that for every $n \in \mathbb {N}$ and any complex numbers $\left\{ c_{k}\right\}_{1}^{n}$ the inequality

   $$\gamma _{1}\sum _{k = 1}^{n}\left|c_{k}\right|^{2} \leqslant \left\| \sum _{k = 1}^{n} c_{k} x_{k}\right\| ^{2} \leqslant \gamma _{2}\sum _{k = 1}^{n}\left|c_{k}\right|^{2}$$

   holds;

4. the system $\left\{ x_{k}\right\}_{1}^{\infty }$ is complete and minimal and there exist constants $\gamma_{1}, \gamma_{2} > 0$ such that for every $x \in H$

   $$\gamma _{1}\left\| x\right\| ^{2} \leqslant \sum _{k = 1}^{\infty }\left|(x, x_{k})\right|^{2} \leqslant \gamma _{2}\left\| x\right\| ^{2};$$

5. (N. K. Bari, 1951) the system $\left\{ x_{k}\right\}$ is complete and minimal in $H$ (so that it has a unique biorthogonal system $\left\{ y_{k}\right\}$) and for every $x \in H$ the inequalities $\sum_{k = 1}^{\infty }\left|(x, y_{k})\right|^{2} < \infty$ and $\sum_{k = 1}^{\infty }\left|(x, x_{k})\right|^{2} < \infty$ hold;

6. (N. K. Bari, 1951) the system $\left\{ x_{k}\right\}_{1}^{\infty }$ is complete and its Gram matrix $\left((x_{i}, x_{j})\right)_{i,j = 1}^{\infty }$ generates a bounded and invertible operator in the space $l_{2}$;

7. the system $\left\{ x_{k}\right\}$ is an unconditional Schauder basis and is almost normalized, i.e., there exist constants $\gamma_{1}, \gamma_{2} > 0$ such that $\gamma_{1} \leqslant \left\| x_{k}\right\| \leqslant \gamma_{2}, k = 1, 2, \ldots$

(When the constants $\gamma_{1}$ and $\gamma_{2}$ in (3) both equal 1, the system becomes an orthonormal basis; compare with Theorem 3.2.)

Theorem 3.2 (F. Riesz, 1907; E. Fischer, 1907): Let $\left\{ e_{n}\right\}_{1}^{\infty }$ be an orthonormal system in a Hilbert space $H$. Then the following conditions are equivalent: (1) the system $\left\{ e_{n}\right\}_{1}^{\infty }$ is total (i.e., every element of $H$ can be approximated arbitrarily well by finite linear combinations of the $e_{n}$); (2) if $\left(x, e_{n}\right) = 0$ for all $n \in \mathbb {N}$ for some vector $x \in H$, then $x = 0$ (such a system is called complete); (3) $\sum_{n = 1}^{\infty }\left|\left(x, e_{n}\right)\right|^{2} = \left\| x\right\|^{2}$ for every $x \in H$ (Parseval's equality holds); (4) every vector $x \in H$ is uniquely represented as a series in the system $\left\{ e_{n}\right\}_{1}^{\infty }$: $x = \sum_{n = 1}^{\infty }\left(x, e_{n}\right) e_{n}$. An orthonormal system with properties (1)--(4) is called an orthonormal basis.

Let the system $\left\{ x_{n}\right\}_{1}^{\infty }$ form a Riesz basis in a Hilbert space $H$. Prove that the biorthogonal system for it is also a Riesz basis.

Prove that the system $\left\{ \sin^{2}(n t)\right\}_{1}^{\infty }$ does not form a Riesz basis in the space $L_{2}[0, \pi / 2]$.

Prove that the system $\left\{ \sin^{2}(n t)\right\}_{1}^{\infty } \cup \left\{ 1\right\}$ forms a Riesz basis in the space $L_{2}[0, \pi / 2]$.

Prove that the system $\left\{ \sin (2 n t), t \cos (2 n t)\right\}_{0}^{\infty }$ forms a Riesz basis in the space $L_{2}[0, \pi ]$.

(M. I. Kadets, 1964). Let $\left\{ \lambda_{n}\right\}_{1}^{\infty }$ be a sequence of real numbers with the property $\left|\lambda_{n}-n\right| < 1 / 4$ for every $n \in \mathbb {Z}$. Prove that the system $\left\{ e^{i n t}\right\}_{n \in \mathbb {Z}}$ is a Riesz basis in the space $L_{2}[0, \pi ]$.

(K. I. Babenko, 1948). Prove that the system $\left\{ \left|t\right|^{\alpha } e^{i \pi n t}\right\}_{-\infty }^{\infty }$ is a Schauder basis, but is not a Riesz basis, in the space $L_{2}[-1,1]$, if $\left|\alpha \right| < 1 / 2, \alpha \neq 0$.

Prove that the system of Chebyshev polynomials $T_{n}(t) = \cos (n \arccos t), n = 0,1, \ldots$, does not form a Riesz basis in the space $L_{2}[-1,1]$.

Let $p(t) \geqslant 0$ be a summable function on the segment $[0,1]$. In the Hilbert space $L_{2}([0,1], p(t) d t)$, consider an arbitrary orthogonal basis $\left\{ e_{n}\right\}_{1}^{\infty }$. Prove that this system forms a Riesz basis in the space $L_{2}[0,1]$ (with the usual Lebesgue measure) if and only if $\underset {t \in [0,1]}{\operatorname {ess} \sup } p(t) < \infty$ and $\underset {t \in [0,1]}{\operatorname {ess} \inf } p(t) > 0$.

Let $\left\{ x_{n}\right\}_{1}^{\infty }$ be a Riesz basis in the space $L_{2}\left(M_{1}, \mu_{1}\right)$, and $\left\{ y_{n}\right\}_{1}^{\infty }$ a Riesz basis in the space $L_{2}\left(M_{2}, \mu_{2}\right)$. Prove that the system $\left\{ x_{n} \cdot y_{m}\right\}_{m, n = 1}^{\infty }$ forms a Riesz basis in $L_{2}\left(M_{1} \times M_{2}, \mu_{1} \times \mu_{2}\right)$ (cf. problem 3.66).

Let $\left\{ e_{n}\right\}_{1}^{\infty }$ be an orthonormal basis in a Hilbert space $H$. Let $\left\{ x_{n}\right\}_{1}^{\infty }$ be a system of vectors in $H$ for which $\sum_{n = 1}^{\infty }\left\| x_{n}-e_{n}\right\|^{2} < 1$. Prove that the system $\left\{ x_{n}\right\}_{1}^{\infty }$ is a Riesz basis.