In many texts on signal processing, the following notation is used to describe the Fourier transform of a discrete time signal $x$:
$$ \hat{X}\left(e^{j\omega}\right) = \sum\limits_{n=-\infty}^{+\infty} x[n] e^{-j \omega n} $$
I don't understand what the notation $ \hat{X}\left(e^{j\omega}\right)$ is trying to express, or what it even means. Is it a composition of some sort?
The mathematical object $\hat X$ is a function of frequency, why is its argument $e^{j\omega}$ rather than just $\omega$? For instance, is the evaluation of $\hat X (2)$ equal to:
$$\sum\limits_{n=-\infty}^{+\infty} x[n] e^{-j 2 n}$$
or
$$\sum\limits_{n=-\infty}^{+\infty} x[n] 2$$