= \bigl\{f\in{\operatorname{map}}(G,k^n) \mid \langle x, f\rangle=0~\forall x\in W\bigr\}.$$ If $M$ is a finitely generated $kG$-module and $f\colon kG^n\twoheadrightarrow M$ is a left $kG$-surjection, then $f^* \colon M^* \hookrightarrow
{\operatorname{map}}(G,k^n)$ is a right $kG$-injection and $${\operatorname{im}}(f^\ast)=\ker(f)^\perp\subseteq{\operatorname{map}}(G,k^n).$$

Approximation for finitely presented $kG$-modules for Elek’s dimension function {#sec:Approximation_for_finitely_presented_kG-modules_for_Eleks_dimension_function}
===============================================================================

The main result of this section is:

\[the:Approximation\_for\_finitely\_presented\_kG-modules\_for\_Eleks\_dimension\_function\] Let $G$ be a finitely generated amenable group. Consider a sequence of normal subgroups of finite index $$G= G_0 \supseteq G_1 \supseteq G_2 \supseteq \cdots$$ such that $\bigcap_{n \ge 0} G_n = \{1\}$. Then every finitely presented $kG$-module $M$ satisfies $$\dim_{kG}^{{\operatorname{Elek}}}(M)= \lim_{n \to \infty} \frac{\dim_k\bigl(k
  \otimes_{kG_n} M\bigr)}{[G:G_n]}.$$

Its proof needs some preparation.

Throughout, let $G$ be a finitely generated amenable group. For any subset $S\subset G$ let $k[S]$ be the $k$-subspace of $kG$ generated by $S\subset kG$. Let $j[S]\colon k[S]\to k[G]$ be the inclusion and ${\operatorname{pr}}[S]\colon kG\to k[S]$ be the projection given by $${\operatorname{pr}}[S](g)=\begin{cases}
  g & \text{if $g\in S$;}\\
  0 & \text{if $g\in G\backslash S$.}
\end{cases}$$

\[the:equality\_elek\_and\_approximative\_dimension\_of\_finitely\_presented\_modules\] Let $G$ be a finitely generated amenable group. Let $M$ be a finitely presented left $kG$-module $M$ with a presentation $kG^r\xrightarrow{f}kG^s\xrightarrow{p} M\to 0$. For every subset $S\subset G$ we define $$M[S]={\operatorname{coker}}\Bigl({\operatorname{pr}}[S]\circ f\circ j[S]\colon k[S]^r\to
  k[S]^s\Bigr).$$ Let $(F_n)_{n \ge 0}$ 