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Every bounded linear functional on $ℓ^{2}$ is summation against a unique $ℓ^{2}$ sequence

Example

Every bounded linear functional $Λ : ℓ^{2} \to R$ has a unique sequence $b = (b_{n}) \in ℓ^{2}$ such that $Λ (a) = \sum_{n = 0}^{\infty} a_{n} b_{n} (a \in ℓ^{2}),$ and $∥ Λ ∥ = ∥ b ∥_{ℓ^{2}} .$

Facts & Assumptions

Given: A bounded linear functional $Λ$ on $ℓ^{2}$ .

[L1]

The counting-measure corollary says that every bounded linear functional on $ℓ^{p}$ is represented by a unique $ℓ^{q}$ sequence, with equality of norms (Counting measure specializes the representation theorem to $ℓ^{p}$ and $ℓ^{q}$ ).

[L2]

On counting measure over $N$ , the integral pairing is exactly the series pairing for sequences ( $ℓ^{p}$ is the $L^{p}$ space of counting measure).

Verification

technique · Specialize the counting-measure duality corollary to $p=q=2$ and rewrite the integral pairing as an ordinary series

1.1L1given

Apply [L1] with $p = q = 2$ . Then there is a unique sequence $b \in ℓ^{2}$ such that $Λ (a) = \sum_{n = 0}^{\infty} a_{n} b_{n}$ for every $a \in ℓ^{2}$ , and $∥ Λ ∥ = ∥ b ∥_{ℓ^{2}} .$

2.1L2step 1.1∎

The summation formula is exactly the counting-measure integral pairing from [L2], so this is the concrete $ℓ^{2}$ instance of the A-page theorem.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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 B. Folland, Real Analysis, 2nd ed., Section 6.2 (standard reference, not scraped)
Richard F. Bass, Real Analysis for Graduate Students, Section 15.4 (standard reference, not scraped)

Every bounded linear functional on ℓ2ℓ2 is summation against a unique ℓ2ℓ2 sequence