What is the relationship between $E[X|Y=y]$ and $E[X|Y]$?

Question

What is the relationship between $E[X|Y=y]$ and $\space E[X|Y]$, where $(X,Y)$ is a bivariate random variable?

Is it that $$E[E[X|Y=y]] = E[X|Y]?$$

Answer 1

$E[X|Y=y]$ is a number (expected value of $X$ given $Y=y$), $E[X|Y]$ is a random variable, function of random variable $Y$; $g(Y)= E[X|Y]$.

$E[X|Y=y]$ is a value of random variable $g(Y)$ when random variable $Y$ takes value $y$. $$E[E[X|Y=y]] \overset{?}{=} E[X|Y]$$ is wrong, because on the left side we have expected value of a number, which is the same number, and on the right side we have random variable.

Answer 2

We can write

$$\mathbb{E}[X\mid Y] = g(Y)$$ for some Borel measurable function $g$. By definition, $$g(y):= \mathbb{E}[X\mid Y = y]$$