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I need to understand how one can think of a function as a vector (in Hilbert space, more specifically) so I can follow along QM texts. I've read this question's answers, but they were uninspiring to me. I've also read this introduction but found it too short to really grasp the concept.

I've been recommended Cohen-Tannoudji's QM book. Do you have any other recommendations or explanations to offer?

My goal is not only to understand the concept, but use it. Also, for the interested, I'm a chemistry major - mathematics is poorly taught to chemistry majors, in my experience.

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    Are you familiar with the definition of a vector space? It helps to change the way you think of vectors rather than changing the way you think of functions. For example, it is perfectly natural to think of an element of $\mathbb R$ as a vector in the infinite-dimensional $\mathbb Q$-vector space $\mathbb R$.2012-05-28
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    I've had a linear algebra course - nothing fancy. I remember having to verify if X or Y were vector spaces using certain criteria, but these are lost to memory. I'm taking a look at wikipedia now.2012-05-28
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    If you are willing to take on a challenge after you review the definition of vector spaces and **vector space isomorphisms**: a)Verify that the set $Funct(X,F)$ of functions from $X$ to a field $F$ is a vector space under pointwise addition and $(\lambda\cdot f)(x)=\lambda f(x)$. b) The set $\prod_{x\in X} F$ of vectors with positions indexed by $X$ is a vector space under pointwise addition and vector scaling. c) Show $Funct(X,F)\cong \prod_{x\in X} F$ by the isomorphism given in my solution below.2012-05-29

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