I want to know how scientists know that the inner product of f and g equal to integration from $$\int_a^b f(x)g(x)\ dx.$$
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$\begingroup$An inner product on a vector space $V$ over $\mathbb R$ is a function $\langle\cdot,\cdot\rangle : V \times V \to \mathbb R$ which satisfies certain axioms, e.g., $\langle v, v\rangle = 0$ iff $v = 0$, $\langle v, v\rangle \geq 0$ for all $v$.
The definition of an inner product on the vector space of let's say continuous functions on $[0,1]$ of
$$\langle f, g \rangle = \int_0^1 f(x)g(x) \ dx$$
works in that it satisfies those axioms.
Now, given that, it turns out we can use the inner product for a very wide range of useful things.
Historically, the definition was probably inspired by analogy with vectors in $\mathbb R^n$, namely
$$\langle (v_1,v_2,...,v_n),(w_1,w_2,...,w_n)\rangle = \sum_{i=1}^n v_iw_i$$
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