Let $X$ be a set equipped with a sequential convergence relation between sequences in $X$ and points of $X$, written $u_k\to u$, such that every constant sequence converges to its constant value. Let

\begin{align*} F:X\to(-\infty,\infty] \end{align*}

be a function. Define the sequential relaxation

\begin{align*} \overline F:X\to[-\infty,\infty] \end{align*}

by

\begin{align*} \overline F(u)=\inf\{\liminf_{k\to\infty}F(u_k):(u_k)_{k=1}^{\infty}\subset X\text{ and }u_k\to u\}. \end{align*}

Then $\overline F(u)\le F(u)$ for every $u\in X$, and

\begin{align*} \inf_{u\in X}\overline F(u)=\inf_{u\in X}F(u), \end{align*}

with the convention that the infimum of the empty set is $+\infty$. Moreover, if $u\in X$, $\overline F(u)=\inf_{v\in X}F(v)$, and $(u_k)_{k=1}^{\infty}\subset X$ is a recovery sequence for $u$, meaning that $u_k\to u$ and $F(u_k)\to\overline F(u)$ in the extended-real sense, then $(u_k)_{k=1}^{\infty}$ is a minimizing sequence for $F$, meaning that $F(u_k)\to\inf_{v\in X}F(v)$ in the extended-real sense.