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Mean value inequality for a differentiable Banach-valued curve

Statement

Let $X$ be a Banach space, $a < b$ , and let $φ : [a, b] \to X$ be continuous on $[a, b]$ and differentiable on $(a, b)$ in the sense of Fréchet derivative between Banach spaces. If there is $C \geq 0$ with $∥ φ^{'} (s) ∥ \leq C$ for all $s \in (a, b)$ , then $∥ φ (b) - φ (a) ∥ \leq C (b - a)$ . In particular, if $φ$ is continuous, differentiable on $(a, b)$ and $φ^{'} = 0$ there, then $φ$ is constant.

Facts & Assumptions

Given: A real or complex Banach space $X$ (Banach space), read as a real vector space for differentiation so that the real-variable Fréchet derivative of Fréchet derivative between Banach spaces applies (the complex case uses the underlying real structure, and complex differentiability is a special case); real numbers $a < b$ , a curve $φ : [a, b] \to X$ continuous on $[a, b]$ and differentiable on $(a, b)$ , and a constant $C \geq 0$ with $∥ φ^{'} (s) ∥ \leq C$ for all $s \in (a, b)$ .

[F1]

At each $s \in (a, b)$ the Fréchet derivative $φ^{'} (s) : R \to X$ is bounded linear and there is a remainder with $∥ φ (s + h) - φ (s) - φ^{'} (s) h ∥ / ∣ h ∣ \to 0$ as $h \to 0$ (Fréchet derivative between Banach spaces).

[F2]

The norm function is continuous on $X$ and satisfies the reverse triangle inequality $∣ ∥ x ∥ - ∥ y ∥ ∣ \leq ∥ x - y ∥$ (The reverse triangle inequality in a normed space).

[F3]

Vector addition and scalar multiplication on $X$ are continuous (Vector addition and scalar multiplication are continuous in a normed space), so limits of sums and scalar multiples may be taken termwise.

Proof

technique · direct, by a maximal-interval argument on the auxiliary function $d(\tau)=\|\varphi(\tau)-\varphi(r)\|-\widetilde C(\tau-r)$

1.1F2F3

Fix $\tilde{C} > C$ and $r \in (a, b)$ , and put $d (τ) : = ∥ φ (τ) - φ (r) ∥ - \tilde{C} (τ - r)$ for $τ \in [r, b]$ . By [F2] and [F3] the function $d$ is continuous, so $S : = {τ \in [r, b] : d (τ) \leq 0}$ is a nonempty closed subset of $[r, b]$ , since $r \in S$ ; being nonempty and bounded above it has a supremum $τ_{0} \in S$ , which is therefore its maximum.

2.1F1F2step 1.1

If $τ_{0} < b$ then $τ_{0} > r$ : if $τ_{0} = r$ , then [F1] at $r$ (legitimate because $r > a$ ) gives $φ (r + h) = φ (r) + φ^{'} (r) h + o (h)$ with $∥ φ^{'} (r) ∥ \leq C$ , so by [F2] $d (r + h) \leq (C - \tilde{C}) h + o (h) < 0$ for all sufficiently small $h > 0$ , contradicting the maximality of $r = τ_{0}$ ; hence $τ_{0} \in (r, b) \subseteq (a, b)$ , where $φ$ is differentiable.

3.1F1F2step 2.1

If $τ_{0} < b$ , then differentiability at $τ_{0}$ gives $φ (τ_{0} + h) = φ (τ_{0}) + φ^{'} (τ_{0}) h + o (h)$ with $∥ φ^{'} (τ_{0}) ∥ \leq C$ , so for small $h > 0$ [F2] gives $d (τ_{0} + h) \leq d (τ_{0}) + (C - \tilde{C}) h + o (h) \leq (C - \tilde{C}) h + o (h) < 0$ , since $d (τ_{0}) \leq 0$ ; then $τ_{0} + h \in S$ , contradicting the maximality of $τ_{0}$ . Hence $τ_{0} = b$ .

4.1step 3.1

At $τ_{0} = b$ the defining inequality of $S$ reads $d (b) \leq 0$ , that is $∥ φ (b) - φ (r) ∥ \leq \tilde{C} (b - r)$ .

5.1F2F3step 4.1

Letting $r ↓ a$ along a sequence: $φ (r) \to φ (a)$ by continuity and [F3], so [F2] gives $∥ φ (b) - φ (r) ∥ \to ∥ φ (b) - φ (a) ∥$ , and $b - r \to b - a$ ; hence $∥ φ (b) - φ (a) ∥ \leq \tilde{C} (b - a)$ .

6.1step 5.1algebra

Since $\tilde{C} > C$ was arbitrary, $∥ φ (b) - φ (a) ∥ \leq C (b - a)$ .

7.1step 6.1algebra∎

If in addition $φ^{'} = 0$ on $(a, b)$ , take $C = 0$ in [step 6.1]; then for every $t \in (a, b]$ the same argument applied to the restriction of $φ$ to $[a, t]$ gives $φ (t) = φ (a)$ , and $φ (a) = φ (a)$ , so $φ$ is constant on $[a, b]$ .

Depends on

Used by

Fundamental theorem of calculus for Banach-valued continuous curves Lemma

Dependency tree · two levels

16 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 Teschl, Partial Differential Equations: From Classical to Modern (2026 author manuscript; complete 392-page archived text) (standard reference, not scraped)
Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer 2011 (complete 614-page text) (standard reference, not scraped)