Quote by Stephen Tashi
A "dramatic" use of the use of a basis in a vector space is expressing a function as a sum of other functions. For example, fourier series, Chebeshev polynomials.

Careful, you need
Schauder for that. On the topic of Fourier, each piano key corresponds to a sine function. The collection of finite sums gives the set of all piano chords.
There is also
Lattice based crypto, but that is discrete so ymmv.