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I am very naive when it comes to advanced mathematics, so consider me a layman. I am currently taking a course in Quantum mechanics and I've been introduced to the concept of functions being infinite dimensional vectors.

We however covered only functions of one variable and the math makes sense to me (the dot product, projecting functions onto the x basis etc. )

But how do these properties generalize to more than variable? What would be the dot product? I guess a double integral.

So, it would be very helpful if someone can generalize concepts in layman's terms or link some web pages that explain this as I couldn't find any.

Thanks

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Imagine a set of functions $\{\phi_1,\phi_2,\cdots \}$, that follow the condition

$$ \int {\rm d}x ~\mu(x)\phi_\alpha(x)\phi_\beta(x) = \delta_{\alpha \beta} \tag{1} $$

This functions are said to be orthogonal w.r.t to $\mu$. In quantum mechanics these type of functions appear everywhere, the reason being Schrodinger's is a Sturm-Liouville problem which naturally yields solutions of the form (1). When your lecturer said 'a function is a vector' (s)he probably meant that a function $\psi(x)$ can be written as

$$ \psi(x) = \sum_{\alpha}c_\alpha\phi_{\alpha} \tag{2} $$

In the same sense of linear algebra, where the coordinates are the $c_\alpha$s and the basis vectors are the $\phi_\alpha$s. A generalization of Eq.(1) to more than one degree of freedom is also possible, the only change is that $x$ now represents a set of numbers (e.g. two angles)

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