Glossary

13.1 Convex analysis in Banach spaces

3 min read•july 22, 2024

Convex sets and functions play a crucial role in Banach spaces. They're the building blocks for many important theorems and applications in functional analysis. Understanding these concepts is key to grasping more advanced topics.

The is a powerful tool for extending linear functionals and separating convex sets. It has far-reaching consequences, including the existence of non-trivial and the characterization of reflexive Banach spaces.

Convex Sets and Functions in Banach Spaces

Convex sets in Banach spaces

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  • contains all points on the line segment connecting any two points within the set
  • Examples of convex sets in Banach spaces include closed balls B[x0,r]={x∈X:∥x−x0∥≤r}B[x_0, r] = \{x \in X: \|x - x_0\| \leq r\}, hyperplanes {x∈X:f(x)=α}\{x \in X: f(x) = \alpha\} where ff is a continuous linear functional and α∈R\alpha \in \mathbb{R}, and halfspaces {x∈X:f(x)≤α}\{x \in X: f(x) \leq \alpha\} or {x∈X:f(x)≥α}\{x \in X: f(x) \geq \alpha\}

Convex functions in Banach spaces

  • satisfies the inequality f((1−t)x+ty)≤(1−t)f(x)+tf(y)f((1-t)x + ty) \leq (1-t)f(x) + tf(y) for any x,y∈Xx, y \in X and t∈[0,1]t \in [0, 1], meaning the line segment connecting any two points on the graph lies above or on the graph
  • Examples of convex functions in Banach spaces include norms f(x)=∥x∥f(x) = \|x\|, continuous linear functionals f(x)=⟨x∗,x⟩f(x) = \langle x^*, x \rangle where x∗∈X∗x^* \in X^* (dual space), and affine functions f(x)=⟨x∗,x⟩+αf(x) = \langle x^*, x \rangle + \alpha where x∗∈X∗x^* \in X^* and α∈R\alpha \in \mathbb{R}

Hahn-Banach Theorem and Its Consequences

Hahn-Banach theorem applications

  • Hahn-Banach theorem (analytic form) extends any continuous linear functional defined on a subspace to the whole space without increasing its norm
  • Hahn-Banach theorem (geometric form) separates any closed and a point outside it by a hyperplane
  • Consequences of the Hahn-Banach theorem:
    1. Normed spaces have enough continuous linear functionals to separate points
    2. The dual space X∗X^* of a XX is non-trivial (X∗≠{0}X^* \neq \{0\})
    3. The bidual space X∗∗X^{**} is isometrically isomorphic to XX for reflexive Banach spaces

Subdifferentials in Banach Spaces

Subdifferentials in Banach spaces

  • of a convex function ff at x∈Xx \in X is the set ∂f(x)={x∗∈X∗:f(y)≥f(x)+⟨x∗,y−x⟩,∀y∈X}\partial f(x) = \{x^* \in X^*: f(y) \geq f(x) + \langle x^*, y-x \rangle, \forall y \in X\}, generalizing the concept of the gradient for non-differentiable convex functions
  • Properties of subdifferentials:
    • If ff is differentiable at xx, then ∂f(x)={∇f(x)}\partial f(x) = \{\nabla f(x)\}
    • If ff is continuous at xx, then ∂f(x)\partial f(x) is a non-empty, convex, and weak* compact subset of X∗X^*
    • The is a monotone operator, satisfying ⟨x∗−y∗,x−y⟩≥0\langle x^* - y^*, x - y \rangle \geq 0 for any x,y∈Xx, y \in X, x∗∈∂f(x)x^* \in \partial f(x), and y∗∈∂f(y)y^* \in \partial f(y)
  • Applications of subdifferentials include optimality conditions (xx is a minimizer of ff if and only if 0∈∂f(x)0 \in \partial f(x)) and characterizing proximal operators prox⁡f(x)=argmin⁡y∈X{f(y)+12∥y−x∥2}\operatorname{prox}_f(x) = \operatorname{argmin}_{y \in X} \{f(y) + \frac{1}{2}\|y-x\|^2\}

Convex Conjugate Functions

Properties of convex conjugates

  • Convex conjugate (or ) of a convex function ff is defined as f∗(x∗)=sup⁡x∈X{⟨x∗,x⟩−f(x)}f^*(x^*) = \sup_{x \in X} \{\langle x^*, x \rangle - f(x)\} and is always convex, even if the original function is not
  • : f(x)+f∗(x∗)≥⟨x∗,x⟩f(x) + f^*(x^*) \geq \langle x^*, x \rangle for all x∈Xx \in X and x∗∈X∗x^* \in X^*
  • Double conjugate: (f∗)∗=f(f^*)^* = f if and only if ff is convex, lower semicontinuous, and proper
  • Subdifferential characterization: x∗∈∂f(x)x^* \in \partial f(x) if and only if f(x)+f∗(x∗)=⟨x∗,x⟩f(x) + f^*(x^*) = \langle x^*, x \rangle

Examples of convex conjugates

  • Indicator function δC(x)={0,x∈C+∞,x∉C\delta_C(x) = \begin{cases} 0, & x \in C \\ +\infty, & x \notin C \end{cases} where CC is a closed convex set has conjugate equal to the support function δC∗(x∗)=sup⁡x∈C⟨x∗,x⟩\delta_C^*(x^*) = \sup_{x \in C} \langle x^*, x \rangle
  • Norm f(x)=∥x∥f(x) = \|x\| has conjugate equal to the indicator function of the unit ball in the dual space, f∗(x∗)=δB[0,1](x∗)f^*(x^*) = \delta_{B[0, 1]}(x^*)
  • Quadratic function f(x)=12∥x∥2f(x) = \frac{1}{2}\|x\|^2 has conjugate that is also a quadratic function, f∗(x∗)=12∥x∗∥2f^*(x^*) = \frac{1}{2}\|x^*\|^2

Key Terms to Review (33)

Affine Hull: The affine hull of a set of points is the smallest affine space that contains the set. It can be visualized as the 'flat' generated by the points, extending infinitely in all directions along the affine combinations of those points. Understanding the affine hull is essential for discussing geometric properties in spaces like Banach spaces, especially when analyzing convex sets and their relationships to linear structures.
Banach space: A Banach space is a complete normed vector space, meaning it is a vector space equipped with a norm such that every Cauchy sequence converges to a limit within the space. This property of completeness is crucial for ensuring the convergence of sequences, which allows for more robust analysis and applications in functional analysis.
Bounded Linear Operator: A bounded linear operator is a linear transformation between two normed spaces that maps bounded sets to bounded sets, ensuring continuity. This means that there exists a constant $C$ such that for every vector $x$ in the domain, the norm of the operator applied to $x$ is less than or equal to $C$ times the norm of $x$. Bounded linear operators play a crucial role in functional analysis as they preserve structure and facilitate the study of continuity, adjointness, and compactness.
Convex function: A convex function is a type of mathematical function where, for any two points on the graph, the line segment connecting them lies above or on the graph itself. This property means that the function exhibits a 'bowl-like' shape, making it easier to analyze in optimization problems, especially within the context of Banach spaces, where the geometric and topological properties play a significant role.
Convex Hull: The convex hull of a set of points is the smallest convex set that contains all the points. It can be visualized as the shape formed by stretching a rubber band around the outermost points in a set, effectively creating a 'tight' boundary. This concept is fundamental in convex analysis, particularly in studying properties of convex sets and functions within Banach spaces.
Convex Set: A convex set is a subset of a vector space such that for any two points within the set, the line segment connecting those points lies entirely within the set. This property makes convex sets crucial in various areas of mathematics, including optimization and functional analysis, particularly in Banach spaces where the structure of the space plays a key role in understanding functional properties.
Convex set: A convex set is a subset of a vector space such that for any two points within the set, the line segment connecting them lies entirely within the set. This property ensures that if you take any two points in the set and draw a line between them, every point on that line will also be included in the set. Convex sets are fundamental in functional analysis, especially in the study of optimization problems and functional spaces.
Dom(f): The term 'dom(f)' refers to the domain of a function 'f', which is the set of all input values for which the function is defined. In the context of convex analysis in Banach spaces, understanding the domain is crucial as it helps identify the inputs that yield valid outputs for various convex functions. The properties of the domain can significantly influence the behavior of optimization problems, stability analysis, and other aspects of functional analysis.
Dual cone: A dual cone is the set of all continuous linear functionals that are non-negative on a given cone in a vector space. It plays a crucial role in convex analysis as it helps characterize the properties of the original cone, particularly in Banach spaces, where the structure of these cones is significant for optimization and functional relationships.
Dual Optimization: Dual optimization refers to the process of solving a dual problem that is derived from a primal optimization problem, where the goal is to find the best possible solution by maximizing or minimizing a certain objective function subject to constraints. This concept is crucial in understanding the relationship between the primal and dual problems, particularly in convex analysis within Banach spaces, as it provides insight into the structure of optimization problems and their solutions.
Dual Spaces: Dual spaces are a fundamental concept in functional analysis that refer to the space of all continuous linear functionals defined on a given vector space. In the context of Banach spaces, the dual space is particularly important as it captures the notion of linear functionals that behave well under the topology induced by the norm, allowing for deeper insights into the structure and properties of the original space.
Epi(f): The notation epi(f) refers to the epigraph of a function f, which is the set of points lying on or above the graph of f. This concept is vital in convex analysis as it helps in understanding the properties of convex functions, optimization problems, and differentiability within the framework of Banach spaces.
Extreme Point: An extreme point of a convex set is a point that cannot be expressed as a convex combination of other points in the set. In simpler terms, it is a 'corner' or 'edge' point of the set where you cannot find other points within the set that average to it. Understanding extreme points is essential for exploring various properties of convex sets and functions, especially in the context of optimization and functional analysis.
Fenchel Conjugate: The Fenchel conjugate, also known as the convex conjugate, of a function is a fundamental concept in convex analysis, defined for a proper, lower semicontinuous convex function. It transforms a function into another function that captures the dual relationship between the original function and the space in which it is defined. This transformation highlights important properties of convex functions, such as their subgradients and optimality conditions, making it essential in various optimization problems.
Fenchel-Young Inequality: The Fenchel-Young inequality is a fundamental result in convex analysis that relates the convex function of a point to the value of that function at another point, establishing a bound on their difference in terms of the inner product. This inequality highlights the relationship between convex functions and their subgradients, emphasizing the importance of differentiability and the role of duality in optimization problems.
Hahn-Banach Theorem: The Hahn-Banach Theorem is a fundamental result in functional analysis that allows the extension of bounded linear functionals defined on a subspace to the entire space without increasing their norm. This theorem is crucial for understanding dual spaces, as it provides a way to construct continuous linear functionals, which are essential in various applications across different mathematical domains.
Jensen's Inequality: Jensen's Inequality is a fundamental result in convex analysis that states if a function is convex and X is a random variable, then the function's value at the expected value of X is less than or equal to the expected value of the function applied to X. This inequality highlights the relationship between convex functions and expectations, making it an essential tool in various areas of mathematics, including probability theory and optimization in Banach spaces.
L. H. Loomis: L. H. Loomis was a prominent mathematician known for his contributions to functional analysis and convex analysis, particularly in the context of Banach spaces. His work laid the groundwork for understanding the properties of convex sets and functions, as well as the geometrical aspects of functional spaces.
Lagrange Multiplier: The Lagrange multiplier is a mathematical tool used to find the local maxima and minima of a function subject to equality constraints. It helps in optimizing a function by introducing an auxiliary variable that incorporates the constraint into the optimization problem, allowing for solutions in higher-dimensional spaces, such as Banach spaces, particularly when analyzing convex functions.
Lower semicontinuity: Lower semicontinuity refers to a property of a function where the value of the function at a point is less than or equal to the limit inferior of the function values at nearby points. This concept plays a crucial role in analysis, particularly in optimization and convex analysis, as it helps to understand the behavior of functions at their boundaries and the nature of minimizers in Banach spaces.
Minkowski Inequality: The Minkowski inequality is a fundamental result in functional analysis that extends the triangle inequality to L^p spaces. It states that for any two measurable functions, the p-norm of their sum is less than or equal to the sum of their individual p-norms, provided that p \\geq 1. This inequality highlights the importance of norms in vector spaces and is crucial in understanding the structure of Banach spaces.
Normed Space: A normed space is a vector space equipped with a function called a norm that assigns a non-negative length or size to each vector in the space. This norm allows for the measurement of distance and the exploration of convergence, continuity, and other properties within the space, facilitating the analysis of linear functionals, dual spaces, and other important concepts in functional analysis.
Optimality Condition: An optimality condition is a mathematical criterion that determines whether a given point is a local optimum for a function, particularly in the context of optimization problems. These conditions provide necessary or sufficient criteria that a solution must satisfy for it to be considered optimal, guiding the search for minima or maxima of functions defined on convex sets, especially in Banach spaces.
Optimization Problems: Optimization problems involve finding the best solution from a set of possible options, typically maximizing or minimizing a specific objective function while satisfying certain constraints. They are fundamental in various fields, including economics, engineering, and mathematics, and often utilize concepts from geometry, convex analysis, and functional analysis to identify optimal solutions.
Polyhedral Convex Set: A polyhedral convex set is a subset of a vector space that can be defined as the intersection of a finite number of half-spaces, making it a convex shape with flat sides. This means that any line segment connecting two points within the set lies entirely inside it, and the flat surfaces or 'faces' are formed by the boundaries defined by linear inequalities. These sets play an important role in optimization, as they provide a structured way to analyze feasible regions in linear programming.
Proper Convex Function: A proper convex function is a convex function defined on a convex set that is not identically infinite and has at least one point in its domain where it takes finite values. Proper convex functions are important in optimization and variational analysis, as they guarantee the existence of minimizers and help establish foundational concepts in convex analysis.
R. Tyrrell Rockafellar: R. Tyrrell Rockafellar is a renowned mathematician recognized for his pivotal contributions to convex analysis, particularly in the context of optimization theory and variational analysis. His work laid the foundation for much of modern convex analysis in Banach spaces, where he established essential results concerning convex functions, duality, and subdifferentials, significantly impacting mathematical optimization and economic theory.
Separation Theorem: The Separation Theorem is a fundamental concept in convex analysis that provides conditions under which two disjoint convex sets can be separated by a hyperplane. This theorem not only establishes the existence of a separating hyperplane but also offers insights into the properties of convex sets, their geometrical structure, and their relationships within Banach spaces.
Strong Convergence: Strong convergence refers to a type of convergence in a normed space where a sequence of elements converges to a limit in the sense that the norm of the difference between the sequence and the limit approaches zero. This concept is crucial when dealing with bounded linear operators, as it ensures stability and continuity in various mathematical settings, including Banach spaces and Hilbert spaces.
Subdifferential: The subdifferential is a set of generalized derivatives associated with a convex function at a given point, allowing for the characterization of non-differentiable points. It captures all possible slopes of secant lines to the graph of the function, providing a way to analyze functions that may not be smooth. This concept is crucial in convex analysis, particularly in the context of optimization and understanding the behavior of convex functions in Banach spaces.
Subdifferential: The subdifferential is a set of all subgradients at a given point in a convex function, which provides a generalized notion of the derivative in the context of convex analysis. In Banach spaces, this concept is crucial for understanding how convex functions behave and for optimization problems where traditional derivatives may not exist. The subdifferential helps identify points of non-differentiability while still capturing essential directional information about the function's growth.
Supporting Hyperplane Theorem: The Supporting Hyperplane Theorem states that for any convex set in a finite-dimensional vector space, if a point lies outside the convex set, there exists a hyperplane that separates the point from the set. This theorem is crucial in understanding the geometry of convex sets and provides tools for optimization problems, particularly in Banach spaces.
Weak Convergence: Weak convergence refers to a type of convergence in a topological vector space where a sequence converges to a limit if it converges with respect to every continuous linear functional. This concept is crucial for understanding the behavior of sequences in various mathematical structures, particularly in the context of functional analysis and applications in areas like differential equations and optimization.
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