Infinite dimensional vector spaces and how they’re used in quantum mechanics.

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A vector has magnitude and direction, right? So would an infinite-dimensional vector space be pointing in infinitely many directions? Also how are they used in quantum mechanics?

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This is a bit more than ELI5, but hopefully it’ll serve as some sort of explanation.

So the “classical” intepretation of a vector is as you say: an object with a magnitude and direction. However, as you get deeper into math, this concept is generalized and you simply define a vector as an element of a [vector space](https://en.wikipedia.org/wiki/Vector_space#Definition_and_basic_properties). A vector space can be described as a set whose elements fulfill some basic properties (see the link above). Now, [any set of real-valued functions form a vector space](https://en.wikipedia.org/wiki/Vector_space#Function_spaces). These spaces are also in general infinite-dimensional. A special type of function spaces are the so called [L^2 -spaces] (https://en.wikipedia.org/wiki/Lp_space) (if you want to be strict you should note that the elements of L^2 -spaces aren’t really functions, but rather equivalence classes, but that’s besides the point). These spaces are very important in quantum mechanics as they form what’s called a [Hilbert space](https://en.wikipedia.org/wiki/Hilbert_space), which means that you have some sort of geometric interpretation of the space in terms of angles, distances, etc. (yes that’s right, a geometric interpretation of a space of functions. How neat is that?). To read more about this discussion if refer to a thread like [this](https://physics.stackexchange.com/questions/41719/why-we-use-l-2-space-in-qm).