Conventional symbol used for equating functions?

Question

What is the conventional symbol used to express that two functions (that have the same domain and the same codomain) are the same? Is the normal equality symbol used (=), or something else, like the equivalence symbol ($\equiv$)?

As an example; would you write:

$\phi: GL_2 \rightarrow GL_2,\ \ \psi: GL_2 \rightarrow GL_2, \ \ \phi \equiv \psi$

or:

$\phi: GL_2 \rightarrow GL_2,\ \ \psi: GL_2 \rightarrow GL_2, \ \ \phi = \psi$

Answer 1

If you only use the function names, then $\phi=\psi$ ought to be enough. If you instead supply an argument (an element $x\in GL_2$) and write $\phi(x)=\psi(x)$, I would say that you ought to write either "$\phi(x)=\psi(x)$ for all $x$" or "$\phi(x)\equiv \psi(x)$".

However, I don't think there are any strict conventions here. Any way you can make it clear that it's an equality and not an equation should be good enough.

Answer 2

If they agree on all elements, then you usually put equality $=$. Remember that a function can be seen as a vector of elements (at every coordinate you have the corresponding value), so agreement on all coordinates means equality as vectors.