So I have usually seen Fourier coefficients $\hat{f}(n)$ of a $2\pi$ periodic function $f$ defined as:
$$\hat{f}(n) = \frac{1}{2\pi}\int_{-\pi}^{\pi} f(x) \exp (inx) \mathrm{d}\mu(x) $$ So $f$ has series representation:
$$ f(x) = \sum_{n \in \mathbb{Z}} \hat{f}(n)\exp(-inx) $$
Now, my question is: what difference does the change in signs make? I know the point is to leverage the identity:
$$ \langle\exp(-inx) , \exp(ikx) \rangle_{L^2(-\pi,\pi)} = \frac{1}{2\pi} \int_{-\pi}^{\pi} \exp(-inx)\exp(ikx)\mathrm{d}\mu(x) = \delta_{n,k} $$ But do we need the difference in signs? I thought the that the $\exp(inx)$ already form an orthonormal basis of $L^2$, so the above identity should follow without the need for different signs. Also, I had one more question, (albeit basic) but something I still need clarification on. Does the fourier series of every $L^2(-\pi,\pi)$ function converge to the function in norm? I.e, does: $$ \left \|\sum_{|n|<N} \hat{f}(n)\exp(-inx) -f\right \|_{L^2} \to 0 \quad \text{as } N \to \infty$$