LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

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Coordinate vectors form the Hilbert basis of real ell-two

Statement

Let $H = ℓ^{2} (N, R)$ with its canonical real inner product, and for each $j \in N$ let $e_{j}$ be the sequence whose $j$ th coordinate is $1$ and whose other coordinates are $0$ . Then $(e_{j})_{j \in N}$ is a complete orthonormal basis of $H$ . For every $x = (x_{j}) \in H$ , the truncations $\sum_{j < N} x_{j} e_{j}$ converge in $H$ to $x$ as $N \to \infty$ (the $N = 0$ sum is empty and equals $0$ ), and $∥ x ∥_{2}^{2} = \sum_{j \in N} ∣ x_{j} ∣^{2}$ .

Statement references

the canonical square-sum norm and finite-tail property on real sequence space

an orthonormal basis is an orthonormal family with dense linear span

real ell-two is complete for its canonical norm

Teschl, §2.1, full notes

Bühler and Salamon, Definition 2.62 and Exercise 2.64

Facts & Assumptions

Given: The real sequence Hilbert space $H = ℓ^{2} (N, R)$ and the coordinate vectors $(e_{j})$ .

[F1]

Square summability means $\sum_{j} ∣ x_{j} ∣^{2} < \infty$ , the norm squared is that sum, and finite tails of the sum can be made arbitrarily small. Square-summable families on an arbitrary index set and the space $ℓ^{2} (I)$

[F2]

The canonical inner product makes real $ℓ^{2} (N)$ a complete real Hilbert space. The real square-summable sequence space is complete

[F3]

An orthonormal family is a Hilbert basis when its closed linear span is the whole Hilbert space. Orthonormal families, complete orthonormal systems and Hilbert bases

Proof

Given: $H = ℓ^{2} (N, R)$ and $e_{j}$ with coordinate $1$ at $j$ and $0$ elsewhere.

1.1F1F2givenalgebra

For all $i, j$ , the canonical inner product gives $⟨ e_{i}, e_{j} ⟩ = δ_{i j}$ , so the coordinate family is orthonormal.

2.1F1F3step 1.1givenalgebra∎

Let $x = (x_{j}) \in H$ and $ε > 0$ . By [F1], a finite set of coordinates has complement square-sum less than $ε^{2}$ . Every finite subset of $N$ is contained in ${j : j < N}$ for some $N \in N$ , so choose $N$ such that $\sum_{j \geq N} ∣ x_{j} ∣^{2} < ε^{2}$ . The coordinate formula gives $∥ x - \sum_{j < N} x_{j} e_{j} ∥_{2}^{2} = \sum_{j \geq N} ∣ x_{j} ∣^{2} < ε^{2}$ . Thus finite linear combinations of the $e_{j}$ are dense in $H$ ; by [F3], the orthonormal family is a complete orthonormal basis. The same tail estimate proves the stated truncation convergence, and the norm identity is [F1].

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Sources

M. Dashti and A. M. Stuart, The Bayesian Approach to Inverse Problems (arXiv:1302.6989) (standard reference, not scraped)
E. Giné and R. Nickl, Mathematical Foundations of Infinite-Dimensional Statistical Models (Cambridge University Press, 2016) (standard reference, not scraped)