Can someone please clarify explicitly what it means to $$span\{^gf; \, g\in G\}$$ where $^gf(h):=f(g.h)$, $G$ is a locally compact group and $f\in L^{1}(G)$ the convolution space of all complex-valued integrable functions on $G$.
Thank you in advance
It's not clear if ${}^g f$ is giving you trouble or $\operatorname{span}$.
You give the formula ${}^gf(h):=f(g.h)$, which means ${}^gf$ is the translate of the function $f$ by the group element $g$, sending $h\in G\mapsto f(g.h)$. (In the case $F=(\mathbb R,+)$, we would have (for example) ${}^g f(x)=1/(1+(x+g)^2)$ if $f(x)=1/(1+x^2)$, and so on.) For a given $f\in L^1(G)$, $$\operatorname{span}\{{}^gf:g\in G\}$$ denotes the smallest subspace of $L^1$ containing all the translates of $f$.