For a finite dimensional normed space, many of the computations can be simplified by the existence of a finite basis, where every vector in the space is a linear combination of some subset of vectors in the basis. It is possible to extend this idea of a basis to infinite dimensional normed spaces in certain situations. Many infinite dimensional spaces contain an infinite sequence of vectors, known as a fundamental sequence, such that any vector in that space can be approximated by linear combinations of vectors belonging to that fundamental sequence. A normed space is called separable if a fundamental sequence exists.
Many infinite dimensional spaces are easily seen to be separable. For instance, the normed space of square integrable functions with the L2 norm on the unit circle in the complex plane is separable, and its fundamental sequence consists of complex exponential functions, or, the Fourier series of the function. This is shown by the Riesz-Fischer Theorem and forms the basis for Fourier analysis.
The space of all infinitely differentiable functions is also separable, and a fundamental sequence is the sequence of all powers of x. This is shown by Taylor series and Weierstrass's theorem.
Other separable spaces include the l2 space of absolute square summable infinite sequences of complex numbers is also obviously separable. The fundamental sequence is clearly the sequence of vectors whose nth number is 1, and all other numbers are zero. The spaces used in wavelet theory are also separable.
One example of a non-separable space is the l∞ space which is the same as l2 except that the norm is ||(a0, a1, a2...)||∞ = sup an. This space can be shown to be not separable. The proof is not trivial though.