Lemmas in analysis

In mathematical analysis and other fields of mathematics, a "lemma" is a preliminary proposition or statement that is proven to aid in the proof of a larger theorem. The term "lemma" comes from the Greek word "lemma," which means "that which is received" or "that which is taken." In effect, results that are designated as lemmas are often foundational results that help establish more complex results.

Auerbach's lemma is a result in functional analysis, specifically in the area concerning the geometry of Banach spaces. It is often used in the study of dual spaces and the properties of linear functionals. The lemma essentially characterizes the relationship between a subspace of a Banach space and its dual.

The Bramble–Hilbert lemma is a result in the mathematical field of numerical analysis and finite element methods. It provides a fundamental estimate that is crucial in the approximation properties of finite element spaces, particularly in the context of solving partial differential equations.

The Calderón–Zygmund lemma is a fundamental result in the theory of singular integrals and is often used in various areas of analysis, including harmonic analysis and partial differential equations. It is named after the mathematicians Alberto Calderón and Anton Zygmund, who made significant contributions to the field.

In differential geometry and calculus, the concepts of closed and exact differential forms are crucial for understanding forms on manifolds, specifically in the context of integration and topology.

Céa's lemma is a result in the field of functional analysis and calculus of variations, particularly in the context of optimal control problems. The lemma is often used to derive estimates for the behavior of solutions to variational problems. In a general sense, Céa's lemma states that under certain conditions, the error in the approximation of a functional can be controlled in terms of the norm of a corresponding linear functional applied to the error of the function.

The Estimation Lemma is a concept used in mathematical analysis, particularly in the context of sequences and series. It is often associated with the estimation of the behavior of functions or sequences under certain conditions. While the term "Estimation Lemma" may not refer to a single, universally recognized theorem, it typically involves methods to estimate bounds or behavior for sequences, series, or integrals.

Grönwall's inequality is an important result in the field of differential equations and analysis, particularly useful for establishing the existence and uniqueness of solutions to differential equations. It provides a way to estimate functions that satisfy certain integral inequalities. There are two common forms of Grönwall's inequality: the integral form and the differential form.

Jordan's Lemma is a result in complex analysis that is particularly useful in evaluating certain types of integrals involving oscillatory functions over infinite intervals. It provides a method for showing that specific integrals vanish under certain conditions, especially when the integrands involve exponential factors.

Lebesgue's lemma is a result in measure theory related to the behavior of measurable functions and their integrals.

Lions–Magenes lemma is a result in the field of functional analysis, particularly in the context of Sobolev spaces and partial differential equations. It provides a crucial tool for establishing the regularity and control of solutions to elliptic and parabolic differential equations. The lemma is typically used to handle boundary value problems, allowing one to obtain estimates of solutions in various norms, which is essential for understanding the existence and uniqueness of solutions as well as their continuity and differentiability properties.

Mazur's Lemma is a result in functional analysis, specifically in the context of convex analysis and the study of weak compactness in Banach spaces. The lemma is named after the Polish mathematician Stanisław Mazur. The central idea of Mazur's Lemma concerns weakly convergent sequences and the nature of the weak closure of convex sets.

The Poincaré lemma is a fundamental result in differential geometry and algebraic topology that pertains to the properties of differential forms on a differentiable manifold.

The Schwarz lemma is a fundamental result in complex analysis that provides important insights into the behavior of holomorphic functions. Specifically, it applies to holomorphic functions defined on the unit disk (the set of complex numbers whose modulus is less than 1).

Spijker's lemma is a result in functional analysis, specifically dealing with the properties of bounded linear operators on Banach spaces. The lemma provides conditions under which certain sequences of bounded linear operators exhibit specific convergence properties. While Spijker's lemma does not have one widely acknowledged statement applicable in all contexts, it typically relates to convergence properties in the context of compact operators or the spectral theory of linear operators.

The Stewart–Walker lemma is a result in the field of differential geometry, particularly in the study of Riemannian manifolds. It is specifically related to the curvature of manifolds and provides conditions under which the curvature tensor can be expressed in terms of the metric tensor and its derivatives. The lemma is often invoked in the context of proving properties about space forms and the relationship between curvature and geometric structures on manifolds.

Weyl's lemma is a result in the theory of partial differential equations, particularly concerning solutions to the Laplace equation. The lemma states that if a function \( u \) is harmonic (i.e.

Wiener's lemma is a result in functional analysis and harmonic analysis, particularly related to the theory of Fourier series and the spaces of functions. It is named after Norbert Wiener, who contributed significantly to the field.