ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedaudited 2026-09-22

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The standard basis of $ℓ^{2} (N)$

Example

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). In $ℓ^{2} (N, F)$ (Square-summable families on an arbitrary index set and the space $ℓ^{2} (I)$ ) let $u_{n}$ be the family that is $1$ at $n$ and $0$ elsewhere. Then $(u_{n})_{n \in N}$ is an orthonormal basis of $ℓ^{2} (N, F)$ (Orthonormal families, complete orthonormal systems and Hilbert bases), and for every $a = (a_{n}) \in ℓ^{2} (N, F)$ the canonical partial sums converge,

$a = \sum_{n = 0}^{\infty} a_{n} u_{n} in ℓ^{2} -norm, ∥ a ∥_{2}^{2} = \sum_{n = 0}^{\infty} ∣ a_{n} ∣^{2} .$

Facts & Assumptions

[A1]

For $a, b \in ℓ^{2} (N, F)$ the pairing is $⟨ a, b ⟩ = \sum_{n \in N} a_{n} \overline{b_{n}}$ (a finite-subset sum), $∥ a ∥_{2}^{2} = \sum_{n \in N} ∣ a_{n} ∣^{2}$ , and for a finite $F$ the difference satisfies $∥ a - a \cdot 1_{F} ∥_{2}^{2} = \sum_{n \notin F} ∣ a_{n} ∣^{2}$ ; if $\sum_{n} ∣ a_{n} ∣^{2} < + \infty$ then for every real $ε > 0$ some finite $F$ has $\sum_{n \notin F} ∣ a_{n} ∣^{2} < ε$ (Square-summable families on an arbitrary index set and the space $ℓ^{2} (I)$ ).

[A2]

$⟨ u_{n}, u_{m} ⟩ = δ_{n m}$ , so the $u_{n}$ have norm one and are pairwise orthogonal; finite orthogonal sums satisfy Pythagoras (Real and complex inner-product spaces and their induced length, Pythagoras and finite orthogonal sums).

[A3]

The scalar field $F$ is complete because every finite-dimensional real or complex normed space is Banach (Every finite-dimensional normed space is Banach). If a Cauchy sequence $(a^{(m)})$ in $ℓ^{2} (N, F)$ has coordinate-wise limits $a_{n}$ , then $a = (a_{n}) \in ℓ^{2}$ and $a^{(m)} \to a$ : for $ε > 0$ choose $M$ with $∥ a^{(m)} - a^{(p)} ∥_{2} \leq ε$ for $m, p \geq M$ ; for each finite $F$ , letting $p \to \infty$ in the finite sums gives $\sum_{n \in F} ∣ a_{n} - a_{n}^{(M)} ∣^{2} \leq ε^{2}$ ; taking the supremum over finite $F$ gives $∥ a - a^{(M)} ∥_{2} \leq ε$ , so $a \in ℓ^{2}$ and all $a^{(m)}$ with $m \geq M$ are within $ε$ of $a$ (Square-summable families on an arbitrary index set and the space $ℓ^{2} (I)$ ).

[A4]

A nondecreasing sequence of reals bounded above converges to the supremum of its range (A nondecreasing sequence bounded above converges to the supremum of its range, and a nonincreasing sequence bounded below to the infimum).

[A5]

A complete orthonormal family expands every vector as the norm limit of its finite-subset partial sums (Fourier expansion in a Hilbert space, Orthonormal families, complete orthonormal systems and Hilbert bases).

Verification

technique · direct

Given: $F \in {R, C}$ and the coordinate vectors $u_{n} \in ℓ^{2} (N, F)$ .

1.1A2

The coordinate vectors are orthonormal: for all $n, m$ one has $⟨ u_{n}, u_{m} ⟩ = δ_{n m}$ because each finite sum has the single surviving term $n = m$ .

1.2A3

The space $ℓ^{2} (N, F)$ is complete: if $(a^{(m)})$ is Cauchy, then for each fixed $n$ the scalars $a_{n}^{(m)}$ form a Cauchy sequence in the complete field $F$ because $∣ a_{n}^{(m)} - a_{n}^{(p)} ∣ \leq ∥ a^{(m)} - a^{(p)} ∥_{2}$ , hence converge to a scalar $a_{n}$ ; by [A3] the family $a = (a_{n})$ lies in $ℓ^{2}$ and is the limit of the sequence.

2.1step 1.1A1

The closed linear span of the coordinate vectors is all of $ℓ^{2} (N, F)$ : for $a \in ℓ^{2}$ and $ε > 0$ , apply [A1] with $ε^{2} > 0$ to obtain a finite $F$ with $\sum_{n \notin F} ∣ a_{n} ∣^{2} < ε^{2}$ . The vector $a \cdot 1_{F} = \sum_{n \in F} a_{n} u_{n}$ lies in the span and satisfies $∥ a - a \cdot 1_{F} ∥_{2} < ε$ ; hence every $a$ is a limit of span elements.

3.1step 1.1step 1.2step 2.1A1A4A5∎

By steps 1.1, 1.2 and 2.1, $ℓ^{2} (N, F)$ is a Hilbert space and $(u_{n})$ is an orthonormal family with dense span, hence a complete orthonormal family, i.e. an orthonormal basis; the expansion $a = \sum_{n} a_{n} u_{n}$ is then the finite-subset expansion, and the canonical partial sums $\sum_{n < N} a_{n} u_{n}$ converge to $a$ because their distance to $a$ is the square root of the omitted tail $\sum_{n \geq N} ∣ a_{n} ∣^{2}$ , while the nondecreasing partial sums $t_{N} = \sum_{n < N} ∣ a_{n} ∣^{2}$ converge to their supremum $∥ a ∥_{2}^{2}$ by [A4], so the tails tend to $0$ ; the norm formula is the definition of $∥ \cdot ∥_{2}$ in [A1].

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Theo Bühler and Dietmar Salamon, Functional Analysis — Example 2.65, p.87 (standard reference, not scraped)
Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §2.1, p.52 (standard reference, not scraped)

The standard basis of ℓ2(N)ℓ2(N)