Proper, coercive and weakly lower semicontinuous extended-real functionals

Setting. Let $X$ be a real normed space (in particular, a Banach space as in Banach space) and let $A\subseteq X$ be a nonempty subset, the admissible set. An extended-real functional on $A$ is a map $I:A\to (-\mathrm{\infty },+\mathrm{\infty }]$; its effective domain is $\mathrm{dom}I=\{u\in A:I(u)<+\mathrm{\infty }\}$. Properness. $I$ is proper if $\mathrm{dom}I\mathrm{\ne }\mathrm{\varnothing }$, equivalently if ${\inf }_{A}I<+\mathrm{\infty }$; a point of $\mathrm{dom}I$ is a finite competitor. Coercivity. $I$ is coercive on $A$ if for every $M\in \mathbb{R}$ there is $R\ge 0$ such that $I(u)>M$ whenever $u\in A$ and $\mathrm{\parallel }u\mathrm{\parallel }\ge R$; equivalently (the form used below) every sublevel set $\{u\in A:I(u)\le \mathrm{\Lambda }\}$, $\mathrm{\Lambda }\in \mathbb{R}$, is bounded. Weak lower semicontinuity. $I$ is weakly sequentially lower semicontinuous at $u\in A$ if $I(u)\le {\mathrm{lim\hspace{0.17em}inf}}_{j}I(u_j)$ for every sequence $(u_j{)}_{j\in \mathbb{N}}\subseteq A$ with $u_j\rightharpoonup u$ (Weak convergence of nets and sequences), and weakly sequentially lower semicontinuous on $A$ if this holds at every $u\in A$. Analogously $I$ is sequentially lower semicontinuous on $A$ if $I(u)\le {\mathrm{lim\hspace{0.17em}inf}}_{j}I(u_j)$ whenever $u_j\to u$ in norm; all infima and limits inferior are taken in $\bar{\mathbb{R}}=[-\mathrm{\infty },+\mathrm{\infty }]$, using the complete extended order of Every subset of $\bar{\mathbb{R}}$ has a least upper bound and a greatest lower bound in $\bar{\mathbb{R}}$, agreeing with the real supremum and infimum on nonempty sets bounded in $\mathbb{R}$. For an extended-real sequence $(a_j)$, set ${\mathrm{lim\hspace{0.17em}inf}}_j{a}_j:={\sup }_N{\inf }_{j\ge N}{a}_j$; this extends the tail formula of Limit superior and limit inferior of a real sequence as ${\inf }_n{\sup }_{k\ge n}{x}_k$ and ${\sup }_n{\inf }_{k\ge n}{x}_k$ in $\bar{\mathbb{R}}$ to sequences that may contain $+\mathrm{\infty }$ (The extended real line $\bar{\mathbb{R}}=\mathbb{R}\cup \{-\mathrm{\infty },+\mathrm{\infty }\}$, its order, and the arithmetic that is left undefined). In particular, the infimum or limit inferior may equal $-\mathrm{\infty }$. Convention. Only the values on $A$ enter these notions, and $I$ is identified with its restriction to $A$; a point of $X\setminus A$ is inadmissible, not a point where $I$ equals $+\mathrm{\infty }$.

• The two forms of coercivity agree. If $I$ is coercive in the divergence form and $\mathrm{\Lambda }\in \mathbb{R}$, applying the definition with $M=\mathrm{\Lambda }$ gives $R\ge 0$ with $I(u)>\mathrm{\Lambda }$ whenever $u\in A$ and $\mathrm{\parallel }u\mathrm{\parallel }\ge R$, so the sublevel set $\{u\in A:I(u)\le \mathrm{\Lambda }\}$ is contained in the bounded set $\{u\in X:\mathrm{\parallel }u\mathrm{\parallel }<R\}$. Conversely, suppose every sublevel set is bounded and let $M\in \mathbb{R}$ be given; the sublevel set $S=\{u\in A:I(u)\le M\}$ is bounded, so there is $R\ge 0$ with $\mathrm{\parallel }u\mathrm{\parallel }\le R$ for all $u\in S$, and every $u\in A$ with $\mathrm{\parallel }u\mathrm{\parallel }\ge R+1$ lies outside $S$, that is, $I(u)>M$ (a value in $(-\mathrm{\infty },+\mathrm{\infty }]$ fails $I(u)\le M$ exactly when it exceeds $M$). This is the sense in which the equivalence is asserted.

• Properness and a finite infimum. If $u\in \mathrm{dom}I$ then ${\inf }_{A}I\le I(u)<+\mathrm{\infty }$, and conversely if ${\inf }_{A}I<+\mathrm{\infty }$ then not every value of $I$ on the nonempty set $A$ is $+\mathrm{\infty }$, so some $u\in A$ satisfies $I(u)<+\mathrm{\infty }$, that is, $u\in \mathrm{dom}I$.

• The sublevel-set form is the one used in the compactness step of the direct method, and the divergence form is the one recorded in the sources ([MA] Definition 2.3, [G] Definition 4.1, [T] Section 13.2). No convexity, continuity or topology on $A$ is assumed by these definitions.