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Nested finite-basis infima decrease to the infimum over their union

Statement

Let $K_{1} \subseteq K_{2} \subseteq \dots$ be nonempty compact feasible sets in a common finite-basis or continuum feasible set $K$ , and let $E : K \to R$ be continuous. Put $m_{j} = \min_{P \in K_{j}} E (P)$ . Then $m_{j}$ is nonincreasing and $\lim_{j \to \infty} m_{j} = \inf_{P \in ⋃_{j} K_{j}} E (P),$ where the limit is allowed to be $- \infty$ . If additionally every $P \in K$ has a sequence $P_{j} \in ⋃_{j} K_{j}$ with $P_{j} \to P$ in the chosen energy topology and $E (P_{j}) \to E (P)$ , then this limit equals $\inf_{P \in K} E (P)$ . In particular, equality with a continuum infimum requires a proved energy-recovery property for the specified functional; nestedness alone does not imply it.

In this statement, infima are taken in the extended real line $\overline{R} : = R \cup {- \infty, + \infty}$ : for a nonempty set of real values, the infimum is its real greatest lower bound when it is bounded below, and is $- \infty$ when it is unbounded below. In the bounded-below case this agrees with Greatest lower bound (infimum).

Facts & Assumptions

Given: A nested sequence of nonempty compact feasible sets, a common continuous energy, and, for the continuum conclusion, the stated energy-recovery property.

[F1]

A continuous real function on a nonempty compact topological space attains its minimum (A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism).

[F2]

Every nonempty set of real values that is bounded below has a real greatest lower bound by Every nonempty set bounded below has an infimum, characterized by Greatest lower bound (infimum). For a set unbounded below, use the extended-real value $- \infty$ as its infimum. In either case the infimum is a lower bound and is greater than or equal to every other extended-real lower bound.

Proof

1.1F1F2givenchoose

Each $m_{j}$ exists by [F1]. If $j \leq k$ , then $K_{j} \subseteq K_{k}$ , so the minimum over the larger set satisfies $m_{k} \leq m_{j}$ . Thus $(m_{j})$ is nonincreasing. Let $m : = \inf {m_{j} : j \geq 1}$ in the extended-real convention of [F2]. If the set of values is bounded below, then for every $ε > 0$ some $j_{0}$ has $m \leq m_{j_{0}} < m + ε$ by the greatest-lower-bound property; for $j \geq j_{0}$ , monotonicity gives $m \leq m_{j} \leq m_{j_{0}} < m + ε$ , so $m_{j} \to m$ . If the set is unbounded below, for every real $R$ some $j_{0}$ has $m_{j_{0}} < R$ , and monotonicity gives $m_{j} < R$ for every $j \geq j_{0}$ , so $m_{j} \to - \infty$ . Thus the extended-real limit exists and equals $m$ .

2.1F1F2step 1.1givenchoose

Set $a = \inf_{P \in \cup_{j} K_{j}} E (P)$ in the extended-real convention of [F2]. Since each minimum is attained by [F1], every $m_{j}$ is a value of $E$ on the union, so $a$ is a lower bound for ${m_{j} : j \geq 1}$ and [F2] gives $a \leq m$ . For every $P \in \cup_{j} K_{j}$ , choose an index $j_{0}$ with $P \in K_{j_{0}}$ . For all $j \geq j_{0}$ , $m_{j} \leq E (P)$ ; since $m$ is a lower bound for all $m_{j}$ , this gives $m \leq E (P)$ . Thus $m$ is an extended-real lower bound for all values of $E$ on the union, so $m \leq a$ by [F2]. Therefore $m = a$ .

3.1F2step 2.1givenchoose∎

Under the additional recovery hypothesis, fix $P \in K$ . If $a = - \infty$ , then $a \leq E (P)$ . Otherwise, for every $ε > 0$ , the convergence $E (P_{j}) \to E (P)$ gives an index $j$ with $E (P_{j}) < E (P) + ε$ . Since $a \leq E (P_{j})$ , we have $a < E (P) + ε$ for every $ε > 0$ , hence $a \leq E (P)$ . Taking the extended-real infimum over $P \in K$ gives $a \leq \inf_{K} E$ . The reverse inequality follows from $\cup_{j} K_{j} \subseteq K$ , so the two extended-real infima are equal.

Depends on

Used by

Nested trial spaces give nonincreasing energies, sometimes with equality Example

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Sources

Eric Cancès and Claude Le Bris, On the convergence of SCF algorithms for the Hartree-Fock equations, ESAIM: Mathematical Modelling and Numerical Analysis 34 (2000), §2.1 (standard reference, not scraped)
Jean-Luc Guermond, MATH 661 Finite Element Methods, Fall 2025, Chapter 47, §47.2 (standard reference, not scraped)