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The weighted trajectory space is complete

Statement

If $E$ is finite-dimensional and complete, then $(C_{η} (E), ∥ \cdot ∥_{η})$ is a Banach space for every $η > 0$ . No choice principle is used.

Facts & Assumptions

Given: A finite-dimensional complete normed space $E$ , $η > 0$ , and the space $C_{η} (E)$ of Weighted trajectory space for a local center-manifold construction.

[F1]

Weighted trajectory space for a local center-manifold construction: $∥ u ∥_{η} = \sup_{t \in R} e^{- η ∣ t ∣} ∥ u (t) ∥$ .

[F2]

$R$ and $R^{n}$ for $n \geq 1$ with the Euclidean metric are complete, componentwise from the Cauchy criterion in $R$ proves completeness of $R$ and of Euclidean $R^{n}$ for $n \geq 1$ . In the present lemma, completeness of $E$ in its specified norm is a hypothesis, including when $E = {0}$ .

Proof

Given: $E$ , $η$ , and $C_{η} (E)$ as above.

1.1F1constructalgebra

Define $J_{η} u (t) : = e^{- η ∣ t ∣} u (t)$ . This is a linear isometry from $C_{η} (E)$ onto $C_{b} (R, E)$ with the supremum norm: its inverse sends $g$ to $u (t) = e^{η ∣ t ∣} g (t)$ , and both maps preserve continuity and the displayed norms.

1.2F2givenchoose

Let $(g_{n})$ be Cauchy in $C_{b} (R, E)$ . For each $t$ , $(g_{n} (t))$ is Cauchy in $E$ , so define $g (t)$ to be its unique limit. Given $ε > 0$ , choose $N$ such that $\sup_{t} ∥ g_{n} (t) - g_{m} (t) ∥ < ε / 3$ for $n, m \geq N$ ; letting $m$ tend to infinity pointwise gives $\sup_{t} ∥ g_{n} (t) - g (t) ∥ \leq ε / 3$ for $n \geq N$ . Thus $g_{n} \to g$ uniformly, which makes $g$ continuous, and the same bound with one fixed $n \geq N$ shows $g$ is bounded. Pointwise uniqueness defines $g$ without choice.

2.1step 1.1step 1.2∎

Therefore $C_{b} (R, E)$ is complete, and the isometry in step 1.1 transfers completeness to $C_{η} (E)$ .

Remarks

This result supplies only completeness of the trajectory space. The contraction used to construct the center manifold is verified separately in A parameterized C5 center manifold exists at a simple Hopf spectral split.

Depends on

Used by

A parameterized C5 center manifold exists at a simple Hopf spectral split Lemma

Dependency tree · two levels

21 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

J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Springer-Verlag, 1976 (standard reference, not scraped)
Yuri A. Kuznetsov, Elements of Applied Bifurcation Theory, 2nd ed., Springer (standard reference, not scraped)