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Hadamard's lemma in separable Hilbert spaces

Arian Bërdëllima

TL;DR

This work extends Hadamard's Lemma from finite-dimensional Euclidean spaces to separable Hilbert spaces by proving that a smooth function $f$ on an open star-convex neighborhood $U\subset\mathcal{H}$ admits the linear representation $f(x)=f(a)+\langle g(x), x-a\rangle$ with a smooth $g$. Building on this, it derives a Taylor-type expansion with a smooth remainder, provides representations of smooth functions under vanishing higher-order derivatives, and develops an infinitesimal analysis framework in Hilbert spaces via dual-number calculus and the map $\Psi$. The results generalize to separable Banach spaces with a Schauder basis, highlighting the broader applicability to infinite-dimensional smooth analysis and nonlinear operator theory. Overall, the paper connects Hadamard's lemma to infinite-dimensional calculus, enabling systematic infinitesimal reasoning in functional-analytic settings.

Abstract

We extend Hadamard's Lemma to the setting of a separable Hilbert space.

Hadamard's lemma in separable Hilbert spaces

TL;DR

This work extends Hadamard's Lemma from finite-dimensional Euclidean spaces to separable Hilbert spaces by proving that a smooth function on an open star-convex neighborhood admits the linear representation with a smooth . Building on this, it derives a Taylor-type expansion with a smooth remainder, provides representations of smooth functions under vanishing higher-order derivatives, and develops an infinitesimal analysis framework in Hilbert spaces via dual-number calculus and the map . The results generalize to separable Banach spaces with a Schauder basis, highlighting the broader applicability to infinite-dimensional smooth analysis and nonlinear operator theory. Overall, the paper connects Hadamard's lemma to infinite-dimensional calculus, enabling systematic infinitesimal reasoning in functional-analytic settings.

Abstract

We extend Hadamard's Lemma to the setting of a separable Hilbert space.

Paper Structure

This paper contains 7 sections, 6 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hadamard's lemma in separable Hilbert spaces
  4. Consequences of Hadamard's Lemma
  5. Taylor series expansion
  6. Certain representations of a smooth function
  7. Infinitesimal analysis

Key Result

Theorem 1

Let $f$ be a smooth, real-valued function defined on an open, star-convex neighborhood $U$ of a point $a\in \mathcal{H}$. Then $f(x)$ can be expressed, for all $x\in U$, in the form: where $g$ is a smooth mapping on $U$.

Theorems & Definitions (13)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Corollary 4
  • Theorem 5
  • proof
  • Theorem 6
  • ...and 3 more