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The continuous delay-history space is Banach

Statement

For every integer $d \geq 1$ and delay $τ > 0$ , the history space $C_{τ} = C ([- τ, 0], R^{d})$ from Retarded delay equations and their history space, equipped with $∥ ϕ ∥_{\infty} = \sup_{s \in [- τ, 0]} ∥ ϕ (s) ∥_{2}$ , is a Banach space.

Facts & Assumptions

Given: $d \geq 1$ , $τ > 0$ , and the history-space definition.

[F1]

The history space is $C ([- τ, 0], R^{d})$ with the supremum of the Euclidean norm (Retarded delay equations and their history space).

[F2]

The Euclidean norm on $R^{d}$ satisfies the norm axioms, including the triangle inequality (Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation).

[F3]

For every nonempty compact interval $J$ and $d \geq 1$ , $C (J, R^{d})$ is complete in the metric $d_{\infty} (x, y) = \sup_{s \in J} ∥ x (s) - y (s) ∥_{2}$ (Continuous $R^{n}$ -valued curves on a nonempty compact interval form a complete supremum-metric space).

[F4]

A real normed space is Banach exactly when its norm-induced metric is complete (Banach space).

Proof

1.1givenF1F3

The interval $J = [- τ, 0]$ is nonempty and compact. By [F1], $C_{τ} = C (J, R^{d})$ , so [F3] gives the finite supremum metric $d_{\infty}$ and its completeness. In particular $∥ ϕ ∥_{\infty} = d_{\infty} (ϕ, 0)$ is finite for every $ϕ \in C_{τ}$ .

2.1F2step 1.1algebra

The Euclidean norm properties in [F2], applied pointwise and then followed by taking suprema, give nonnegativity, definiteness, scalar homogeneity, and the triangle inequality for $∥ \cdot ∥_{\infty}$ . Its induced metric is therefore exactly $d_{\infty}$ from step 1.1.

3.1F4step 1.1step 2.1∎

Step 1.1 gives completeness for this induced metric, and [F4] defines that complete normed space to be Banach. Hence $C_{τ}$ is Banach.

Depends on

Used by

Nothing in the library uses this result yet.

Cited to discharge well-definedness by Retarded delay equations and their history space.

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Sources

M. Zennaro, The Numerical Solution of Delay Differential Equations, Dobbiaco 2006 summer-school lecture notes (standard reference, not scraped)
Nicola Guglielmi and Ernst Hairer, Solving Delay Differential Equations, Chapter 1 (standard reference, not scraped)