# What is a complex inner product space “really”?

To be clear on this, I know what is the definition of an inner product space and some properties and theorems about them. What I am asking for is an intuition for this definition in the complex case. In the real case, the intuition (or at least one of them) is geometric: The inner product of two vectors is the length of the projection of the first to the second scaled by the norms of both vectors so that it is symmetric (modulo some details). In particular I (and everybody else) think of "inner product zero" as geometric orthogonality and of orthonormal bases as, well, orthonormal bases and so on. The question is, what should I think about when working with complex (or should I say hermitian?) inner product spaces? what is the "meaning" of the complex number associated to two vectors called their inner product?

I will be happy to hear all kinds of answers. For example, what physical phenomena does it model or in what mathematical situations does in "naturally" appear. Answers that stress the "nice structure" resulting are also welcome, yet I feel that by itself it is a bit unsatisfying.

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 As for physical phenomena, complete complex inner product spaces (a.k.a. Hilbert spaces) are the common underlying structure of quantum mechanics. – Henning Makholm Nov 2 '11 at 17:43 I am vaguely aware of that, I hoped for a more elementary example, though I would love to hear a more detailed explanation of this. – KotelKanim Nov 2 '11 at 17:46 I have intuition for real inner product spaces of low dimension from Euclidean geometry. Anything else I understand only algebraically. For example the problem of resolving a vector into $n$ linearly independent "components" requires solving $n$ equations in $n$ unknowns in general, but reduces to computing just $n$ inner products if the components are orthogonal. This is clear from the algebra, and so useful it almost justifies arbitrary inner products all by itself. It is good to ask questions like yours, but I wanted to point out that it is not really necessary to have answers to them. – leslie townes Nov 2 '11 at 22:27