Bari's theorem on quadratically close systems

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Problem 3.68 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 3.3 "Orthogonal and orthonormal systems and bases in Hilbert spaces"

(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 $H$ .

Problem 12 from Halmos P.R. "A Hilbert Space Problem Book" [Hal82]More problems in Chapter 1 "Vectors"

If $\left\{ e_1, e_2, e_3, \ldots \right\}$ is an orthonormal basis for a Hilbert space $\mathbf{H}$ , and if $\left\{ f_1, f_2, f_3, \ldots \right\}$ is an orthonormal set in $\mathbf{H}$ such that

\sum _{j=1}^{\infty } \left\| e_j - f_j\right\| ^{2} < \infty ,

then the vectors $f_j$ span $\mathbf{H}$ (and hence form an orthonormal basis for $\mathbf{H}$ ).

Problem 1.51 from Heil C. "A Basis Theory Primer" [Hei11]More problems in Chapter 1 "Banach Spaces and Operator Theory"

(a)(b)

Let $\left\{ x_{n}\right\}$ be an orthonormal basis for a separable Hilbert space $H$ .

(a)

Show that if $\sum \left\| x_{n}-y_{n}\right\|^{2} < 1$ , then $\left\{ y_{n}\right\}$ is complete in $H$ .

(b)

Show that this conclusion need not hold if $\sum \left\| x_{n}-y_{n}\right\|^{2} = 1$ .

Problem 10.6.14 from Petrovic J.S. "Measure and Integral: Theory and Practice" [Pet25]More problems in Section 10.6 "Bessel's Inequality and Orthonormal Bases"

Let $\mathcal{H}$ be Hilbert space, let $\left\{ x_{n}\right\}_{n=1}^{\infty }$ be an orthonormal basis of $\mathcal{H}$ , and let $\left\{ y_{n}\right\}_{n=1}^{\infty }$ be an orthonormal system in $\mathcal{H}$ . Suppose that $\sum_{n} \left\| x_{n}- y_{n} \right\|^{2}<1$ . Prove that $\left\{ y_{n}\right\}_{n=1}^{\infty }$ must also be an orthonormal basis for $\mathcal{H}$ . Is this is still true if we only assume that $\sum_{n}\left\| x_{n}-y_{n}\right\|^{2}<\infty$ ?

Problem 11.2.10 from Petrovic J.S. "Measure and Integral: Theory and Practice" [Pet25]More problems in Section 11.2 "Banach Space Isomorphisms"

Let $\mathcal{H}$ be Hilbert space, let $\left\{ x_{n}\right\}_{n=1}^{\infty }$ be an orthonormal basis of $\mathcal{H}$ , let $0 \leq C<1$ , and let $\left\{ y_{n}\right\}_{n=1}^{\infty }$ be a sequence in $\mathcal{H}$ such that for any scalars $\left\{ \alpha_{n}\right\}$ ,

\left\| \sum _{k=1}^{n} \alpha _{k}\left(x_{k}-y_{k}\right)\right\| \leq C\left\| \sum _{k=1}^{n} \alpha _{k} x_{k}\right\| .

Prove that for each $x \in \mathcal{H}$ there are scalars $\left\{ \beta_{n}\right\}$ such that $x=\sum_{k=1}^{\infty } \beta_{k} y_{k}$ , with the series converging in the norm of $\mathcal{H}$ .

Problem 3.12.5 from Rudin W. "Functional Analysis" [Rud91]More problems in Chapter 3.12 "Bounded Operators on a Hilbert Space"

(a)(b)(c)

Suppose $\left\{ u_{n}\right\}$ is a sequence of unit vectors in $H$ (that is $\left\| u_{n}\right\| = 1$ ), and assume that

\Gamma ^{2} = \sum _{i \neq j}\left|\left(u_{i}, u_{j}\right)\right|^{2} < \infty

If $\left\{ \alpha_{i}\right\}$ is any sequence of scalars, prove that

(1-\Gamma ) \sum _{i = m}^{n}\left|\alpha _{i}\right|^{2} \leq \left\| \sum _{i = m}^{n} \alpha _{i} u_{i}\right\| ^{2} \leq (1+\Gamma ) \sum _{i = m}^{n}\left|\alpha _{i}\right|^{2}

and deduce that the following three properties of $\left\{ \alpha_{i}\right\}$ are equivalent to each other:

(a)

$\sum_{i = 1}^{\infty }\left|\alpha_{i}\right|^{2} < \infty$ .

(b)

$\sum_{i = 1}^{\infty } \alpha_{i} u_{i}$ converges, in the norm of $H$ .

(c)

$\sum_{i = 1}^{\infty } \alpha_{i}\left(u_{i}, y\right)$ converges, for every $y \in H$ .

This generalizes Theorem 12.6.

Problem 6B.6 from Axler S. "Linear Algebra Done Right" [Axl24]More problems in Section 6B "Orthonormal Bases"

(a)(b)

Suppose $e_{1}, \ldots , e_{n}$ is an orthonormal basis of $V$ .

(a)

Prove that if $v_{1}, \ldots , v_{n}$ are vectors in $V$ such that

\left\| e_{k}-v_{k}\right\| < \frac{1}{\sqrt{n}}

for each $k$ , then $v_{1}, \ldots , v_{n}$ is a basis of $V$ .

(b)

Show that there exist $v_{1}, \ldots , v_{n} \in V$ such that

\left\| e_{k}-v_{k}\right\| \leq \frac{1}{\sqrt{n}}

for each $k$ , but $v_{1}, \ldots , v_{n}$ is not linearly independent. This exercise states in (a) that an appropriately small perturbation of an orthonormal basis is a basis. Then (b) shows that the number $1 / \sqrt{n}$ on the right side of the inequality in (a) cannot be improved upon.

References

[БСШ17] Бородин П.А., Савчук А.М., Шейпак И.А. Задачи по функциональному анализу. МЦНМО, 2017.

[Hal82] Halmos P.R. A Hilbert Space Problem Book. 2nd ed., Springer-Verlag, 1982.

[Hei11] Heil C. A Basis Theory Primer. Birkhäuser, 2011.

[Pet25] Petrovic J.S. Measure and Integral: Theory and Practice. CRC Press, 2025.

[Rud91] Rudin W. Functional Analysis. 2nd ed., McGraw-Hill, 1991.

[Axl24] Axler S. Linear Algebra Done Right. 4th ed., Springer, 2024.