I'm reading Halmos' text and he defines 'basis' as a maximal orthonormal subset of a Hilbert space $H$, but this definition seems inconsistent with the standard definition of basis.
With the standard terminology, a maximal orthonormal subset $\beta$ of $H$ is a linearly independent subset but there is no gurantee that $\beta$ spans $H$. Is $\beta$ actually the basis? If not, what is an example of this case?
On
A Hilbert space is separable (i.e. has a countable dense subset) if and only if it has a countable orthonormal basis. If you're considering non-separable Hilbert spaces, it gets a bit hairier; see this mathoverflow question: https://mathoverflow.net/questions/36734/orthonormal-basis-for-non-separable-inner-product-space
In general an orthonormal basis is not a basis in the algebraic sense. You need to use infinite "linear combinations" to get all the vectors in the space, not just finite ones. For example, in the sequence space $\ell^2$, $x = (x_1, x_2, \ldots)$ can be written as $\sum_{i=1}^\infty x_i e_i$ where $e_i$ are the standard unit vectors.