Understanding a definition I'm reading this article on Circuit Theory by John Baez
https://ncatlab.org/johnbaez/show/Circuit+theory,
and I'm having trouble in understanding how a map is defined.  In the &quot;Cochain Complexes From Circuits&quot; section he defines a map $r$ which should be the canonical isomorphism between the Hilbert space of chains $C_i(\Gamma)$ of a circuit and its dual $C^i(\Gamma)$.  He says that this map is given by the relation
$$
a(\beta)=\langle r(a),\beta\rangle,
$$
where $a\in C_i(\Gamma)$ and $\beta\in C^i(\Gamma)$.
What I don't understand is:  What is $a(\beta)$?  That is, how can I evaluate a chain in a cochain?  Moreover, what is the angle bracket between two cochains, as in the second member?
So, what is this function $r$ that he is defining?
I think I've got it.  With $a(\beta)$ he really meant $\beta(a)$, so the relation is
$$
\beta(a)=\langle r(a), \beta\rangle,\quad\text{for each $a\in C_i(\Gamma)$ and $\beta\in C^i(\Gamma)$.}
$$
This is equivalent to saying that $r(b)=\langle b,\cdot\rangle$ for each $b\in C_i(\Gamma)$, which is the standard way of defining an isomorphism given an inner product.
In fact, for each $b\in C_i(\Gamma)$,
$$
\eqalign{
(rb)(a)&amp;=\langle b,a\rangle\cr
&amp;=\langle a,b\rangle\cr
&amp;=\langle r(a),r(b)\rangle.
}
$$
So, upon defining $\beta=rb$, we have $\beta(a)=\langle r(a),\beta\rangle$.
I think I've got it.  With $a(\beta)$ he really meant $\beta(a)$, so the relation is
$$
\beta(a)=\langle r(a), \beta\rangle,\quad\text{for each $a\in C_i(\Gamma)$ and $\beta\in C^i(\Gamma)$.}
$$
This is equivalent to saying that $r(b)=\langle b,\cdot\rangle$ for each $b\in C_i(\Gamma)$, which is the standard way of defining an isomorphism given an inner product.
In fact, for each $b\in C_i(\Gamma)$,
$$
\eqalign{
(rb)(a)&amp;=\langle b,a\rangle\cr
&amp;=\langle a,b\rangle\cr
&amp;=\langle r(a),r(b)\rangle.
}
$$
So, upon defining $\beta=rb$, we have $\beta(a)=\langle r(a),\beta\rangle$.