two questions about character tables

I have learnt about group theory before I learnt MO theory, so I have a mathematical understanding of those orthogonality theorems and character tables. 


I will use the character table of $D\_{4h}$ to ask my questions:
[![enter image description here](https://i.stack.imgur.com/eTiWy.png)](https://i.stack.imgur.com/eTiWy.png)


1. **About basis functions** 


In the character table, we can see that the basis functions of $A\_{1u}$ are not listed. Does it mean that any function can be a basis function of the corresponding invariant subspace? 


For $E\_{u}$, we know that the dimension of the invariant subspace is two, then does it mean that any pair of functions listed can be regarded as a set of basis functions?
2. **About labels of irreducible representations**


I learnt that those labels for irreducible representations are called Mulliken symmetry labels. But I don't understand what are the symmetries that we check? For instance, $A\_{1g}$, is that we check the symmetries of $x^2 + y^2$ or $z^2$ (which are basis functions) that we get such a label?


