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The finite Bessel inequality and best approximation by a finite orthonormal family

Statement

Let $(e_{i})_{i \in I}$ be an orthonormal family in a real or complex inner-product space $H$ (Orthonormal families, complete orthonormal systems and Hilbert bases), let $F \subseteq I$ be finite, and for $x \in H$ put

$P_{F} x : = \sum_{i \in F} ⟨ x, e_{i} ⟩ e_{i} .$ Then: 1. $P_{F} x$ lies in the span of ${e_{i} : i \in F}$ , and $∥ P_{F} x ∥^{2} = \sum_{i \in F} ∣ ⟨ x, e_{i} ⟩ ∣^{2}$ ; 2. the residual $x - P_{F} x$ is orthogonal to every $e_{j}$ with $j \in F$ , hence to every vector of the span of ${e_{i} : i \in F}$ ; 3. $∥ x - P_{F} x ∥^{2} = ∥ x ∥^{2} - \sum_{i \in F} ∣ ⟨ x, e_{i} ⟩ ∣^{2}$ , and therefore the finite Bessel inequality holds: $\sum_{i \in F} ∣ ⟨ x, e_{i} ⟩ ∣^{2} \leq ∥ x ∥^{2};$ 4. $P_{F} x$ is the unique best approximation to $x$ from the span of ${e_{i} : i \in F}$ : for every $y$ in that span, $∥ x - y ∥^{2} = ∥ x - P_{F} x ∥^{2} + ∥ P_{F} x - y ∥^{2} \geq ∥ x - P_{F} x ∥^{2},$ with equality if and only if $y = P_{F} x$ .

At $F = \emptyset$ the sum defining $P_{F} x$ is empty, so $P_{F} x = 0$ and the identities read $∥ x ∥^{2} = ∥ x ∥^{2}$ .

Facts & Assumptions

[A1]

The inner product is linear in the first argument and conjugate-linear in the second, and $∥ v ∥^{2} = ⟨ v, v ⟩$ (Real and complex inner-product spaces and their induced length).

[A2]

$⟨ e_{i}, e_{j} ⟩ = δ_{i j}$ , so in particular $∥ e_{i} ∥ = 1$ and $e_{i} \neq 0$ (Orthonormal families, complete orthonormal systems and Hilbert bases).

[A3]

Orthogonality is symmetry-compatible and means $⟨ u, v ⟩ = 0$ ; every vector is orthogonal to $0$ (Orthogonality and the orthogonal complement).

[A4]

Pairwise orthogonal finite sums satisfy Pythagoras: $∥ \sum_{j} z_{j} ∥^{2} = \sum_{j} ∥ z_{j} ∥^{2}$ when the $z_{j}$ are pairwise orthogonal, the empty sum being $0$ (Pythagoras and finite orthogonal sums).

[A5]

The span of a set of vectors consists of its finite linear combinations and is a linear subspace (Linear subspace of a vector space).

Proof

technique · direct

Given: An orthonormal family $(e_{i})_{i \in I}$ in $H$ , a finite $F \subseteq I$ , a vector $x \in H$ , and $P_{F} x = \sum_{i \in F} ⟨ x, e_{i} ⟩ e_{i}$ .

1.1

The vector $P_{F} x$ is the finite linear combination of the vectors $e_{i}$ , $i \in F$ , with coefficients $⟨ x, e_{i} ⟩$ , so it lies in the span of ${e_{i} : i \in F}$ by [A5]; and $F = \emptyset$ gives $P_{F} x = 0$ with empty sums on both sides of the two norm identities.

A5A1

1.2

For every $j \in F$ , linearity in the first argument and [A2] give $⟨ P_{F} x, e_{j} ⟩ = \sum_{i \in F} ⟨ x, e_{i} ⟩ ⟨ e_{i}, e_{j} ⟩ = ⟨ x, e_{j} ⟩$ , hence $⟨ x - P_{F} x, e_{j} ⟩ = ⟨ x, e_{j} ⟩ - ⟨ P_{F} x, e_{j} ⟩ = 0$ : the residual is orthogonal to every $e_{j}$ with $j \in F$ .

A1A2A3

1.3

Expanding the pairing of $P_{F} x$ with itself, $⟨ P_{F} x, P_{F} x ⟩ = \sum_{i, j \in F} ⟨ x, e_{i} ⟩ \overline{⟨ x, e_{j} ⟩} ⟨ e_{i}, e_{j} ⟩ = \sum_{i \in F} ∣ ⟨ x, e_{i} ⟩ ∣^{2}$ , so $∥ P_{F} x ∥^{2} = \sum_{i \in F} ∣ ⟨ x, e_{i} ⟩ ∣^{2}$ by [A1].

A1A2algebra

2.1

If $y$ lies in the span of ${e_{i} : i \in F}$ , then $y = \sum_{i \in F} λ_{i} e_{i}$ for suitable scalars, so conjugate-linearity in the second argument together with step 1.2 gives $⟨ x - P_{F} x, y ⟩ = \sum_{i \in F} \overline{λ_{i}} ⟨ x - P_{F} x, e_{i} ⟩ = 0$ ; thus the residual is orthogonal to the whole span.

step 1.2A1A5

2.2

The decomposition $x = P_{F} x + (x - P_{F} x)$ has orthogonal summands by step 1.2, so Pythagoras and step 1.3 give $∥ x ∥^{2} = ∥ P_{F} x ∥^{2} + ∥ x - P_{F} x ∥^{2} = \sum_{i \in F} ∣ ⟨ x, e_{i} ⟩ ∣^{2} + ∥ x - P_{F} x ∥^{2}$ ; since $∥ x - P_{F} x ∥^{2} \geq 0$ this yields the difference identity and the finite Bessel inequality.

step 1.2step 1.3A4algebra

3.1

For $y$ in the span, the vector $P_{F} x - y$ also lies in the span, so it is orthogonal to $x - P_{F} x$ by step 2.1; the difference identity of step 2.2 applied to the orthogonal decomposition $x - y = (x - P_{F} x) + (P_{F} x - y)$ gives $∥ x - y ∥^{2} = ∥ x - P_{F} x ∥^{2} + ∥ P_{F} x - y ∥^{2}$ , which is at least $∥ x - P_{F} x ∥^{2}$ and is equal to it exactly when $∥ P_{F} x - y ∥ = 0$ , that is exactly when $y = P_{F} x$ by definiteness of the norm.

step 2.1step 2.2A4A5A1

4.1

Steps 1.1 and 1.3 give the first claim, steps 1.2 and 2.1 the second, step 2.2 the third, and step 3.1 the fourth, so all four assertions hold for every finite $F$ and every $x$ .

step 1.1step 1.3step 2.1step 2.2step 3.1∎

Depends on

Used by

A separable infinite-dimensional Hilbert space is ℓ² Corollary
Trace of a trace class operator Definition
The standard basis of ℓ²(ℕ) Example
Nuclear series characterizes trace norm Lemma
Positive square root of a compact positive operator Lemma
Singular values equal approximation numbers Lemma
A Hilbert space with a dense sequence has a finite or countable orthonormal basis Theorem
A Hilbert space with a given orthonormal basis is ℓ² of the index set Theorem
Cyclicity of the trace Theorem
Fourier expansion in a Hilbert space Theorem
Hilbert Schmidt operators form a two sided ideal Theorem
Hilbert–Schmidt operators are compact Theorem
L two kernels give Hilbert–Schmidt operators Theorem
Parseval equivalences for an orthonormal family Theorem
Singular value decomposition for compact operators Theorem
Spectral theorem for compact self adjoint operators Theorem
The Bessel inequality for an arbitrary orthonormal family Theorem
Trace class iff product of two Hilbert Schmidt operators Theorem
Trace is absolutely convergent and basis independent Theorem

Dependency tree · two levels

15 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §2.1, pp.47–48, Lemma 2.1 (standard reference, not scraped)
Theo Bühler and Dietmar Salamon, Functional Analysis — Theorem 2.65 area, p.87 (standard reference, not scraped)