I was following the derivation of the basic Fourier series using orthogonal function. For the set of orthogonal functions $\{\phi_n\}$, say the function $f$ can be defined as:

$$f(x) = c_0 \phi_0(x) + c_1 \phi_1(x) +....+ c_n \phi_n(x) +.....$$ For constants $c_i$ They have the inner product: $$\langle f, \phi_n \rangle = \int_{-L}^{L} f(x) \phi_n(x) \, dx$$

This is where I get confused. The simplify $$f(x) = \sum_{n = 0}^{\infty} c_n \phi_{n}(x)$$ which they then take out a factor of sum from n= 0 to infinity of $c_n$ which they simplify to $c_n$ and leave $\phi_n(x)$ in. How does the infinite sum become $c_n$?

If I understand your question The infinite sum of cn phi n (x) = cn summation phi n (x) because cn constant take the limit

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