Real life applications of general vector spaces Students familiar with Euclidean space find the introduction of general vectors spaces pretty boring and abstract particularly when describing vector spaces such as set of polynomials or set of continuous functions. Is there a tangible way to introduce this? Are there examples which will have a real impact? I would like to introduce this in an engaging manner to introductory students. Are there any real life applications of general vector spaces?
Many years ago I was having a beer with a couple of fellow math grad students at some place around Harvard Square, and we overheard some guy at the next table trying to impress a girl telling her that he was taking a Linear Algebra course which was "so difficult" having to deal with spaces of "many dimensions".

I am not sure that this Linear Algebra technique of picking up girls at a bar may be listed among "real life applications", but, if it works, sure offers an important motivation.
In many Mathematical problems practical or theoretical we have a Set which may be sequence of numbers, continuous Functions etc. In which addition, subtraction, multiplication and division of two number belongs to that set. This suggests the concept of vector space