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.
Best Answer
Think of a inner product between $v_1$ on $v_2$ as an answer to the question: "How much of $v_2$ can be described using $v_1$".
This intuition corresponds nicely to what you know about orthogonality: if two vectors are orthogonal (inner product $0$), that means you can't describe $v_1$ using a linear combination of vectors that includes $v_2$.
In complex space, this is exactly the same idea - two vectors are orthogonal if one cannot be represented using the second one. For instance in $\mathbb{C^2}$, $v_1=(i,0)$ and $v_2=(0,1)$ are orthogonal for exactly the same reason they $v_1=(1,0)$ and $v_2=(0,1)$ are in $\mathbb{R^2}$.
One common example is the Fourier Series: the Fourier coefficients are a measure of "how much" of a given frequency is in our function. This doesn't change when we move to the complex representation, we just ask "how much" $e^{inx}$ is in $f(x)$ instead of $\sin(nx)$.
A concrete example may help:
An inner product over the space of all square-integrable functions on $[−π, π]$ may be defined : $$\langle f,g \rangle=\int_{-\pi}^{\pi} f(x) g(x)^*dx$$ Now, as an example, take $f(x)=\sin(x), \ g(x) = e^{ix}$. The inner product is: $$\langle f,g \rangle =\frac{1}{2i}$$ This is exactly "how much" of $e^{ix}$ is in $\sin(x)$, since, as you probably know: $$\sin(x) = \frac{e^{ix}}{2i} -\frac{e^{-ix}}{2i}$$