If $\sum_n \ c_n \ \psi_n(x,t)$ represents an arbitrary state for a given solution to the TISE, what are the bases for a free particle? If $\sum_n c_n \psi_n(x,t)$ represents an arbitrary vector in the Hilbert space of solutions to Schrodinger's equation with a given potential function, this makes makes sense to me. Each $\psi_n$ can be thought as a basis vector, and thus each state is a linear combination of basis vectors. However, an arbitrary vector for a free particle is represented as:
$$\psi(x,t) = \int_{-\infty}^{\infty} A(k) \ e^{i(kx-\omega t)} \ dk$$ rather than a sum. I feel like I've lost the ability to think in a linear mathematical way here. Are there no bases? If there are, how could they be? A basis for me assumes an arbitrary vector by a linear combination such as:
$$u \in V = \sum_n \alpha_i v_i, \forall \alpha_i \in F, \forall v_i \in V$$
Where $V$ is a vector space and $F$ is a field. I couldn't think of it in terms of an integral.
 A: The integral is a sum. We just changed from a discrete index to a continuous one, since $k$ isn't bounded by any quantization conditions. The bases are the separatable solutions to the free particle Hamiltonian, but since they aren't normalizable, they can't represent a physical state. Although, we can think of them as handy mathematical idealizations.
A: The basis functions are every possible sine and cosine wave of every possible real-number wavenumber (often called frequency, but that's a time concept) and unit amplitude.
There's a small problem in that basis functions should be normalized, the integral from -inf to +inf being unity.  But over an infinite range, that's a problem. So we cheat. This is a common technique in theoretical physics, so pay attention!  
We imagine a large box, let's say it's LxLxL in size, and then we can integrate over a finite domain.  Then the basis functions aren't having wavenumbers of all possible real numbers, but only integer multiples of some small wavenumber.  The old Particle in a Box problem we all learned in the 2nd week of Quantum Mechanics 101.  Let L increase greatly and keep increasing.  There are more and more valid stationary eigenstates per interval of energy. One speaks of "density of states" meaning how many eigenstates there are in some small delta-E of energy. It varyies with the energy.  It's sorta like computers representing real numbers with more and more bits of precision.  At some point, you can say "good enough, these look like real numbers".   From a formal mathematical standpoint, you need to look at Measure Theory. This is yet another fine example of how Mathematics as a field of study and body of knowledge benefits from the antics of physicists.  
As L approaches infinity, the numbers of states having an energy between any E and E+dE approaches infinity, but each has infinitesimal contribution to the total.   In the end, you're allowed to use any real number as the wavenumber (aka spatial frequency) and you write amplitude as a function of wavenumber, a real function of a real variable.    We just need to remember to include all phases, which we achieve by having sines and cosines. Mix them together with complex amplitudes as you wish.
This is a rich, deep area of mathematical physics.  Things were especially interesting at the point of history in this subject when theoretical physicists realized they'd have to deal with discrete spectrum states with E<0 and  a continuum of states for E>0 for many simple quantum systems. Today, or heck, even four or five decades ago, it became a (mostly) solved problem.  
