Consider the conditional expectation of $x$ given $y$, $$ \mathbb{E}(x | y) $$ where $x \in X$ and $y \in Y$ where $X, Y$ are Hilbert spaces (possibly infinite dimensional).

**Question :**
I am looking for conditions under which
$$
\mathbb{E}(x | \cdot ) : Y \to X
$$
is a continuous function.

I'll give some simple examples to illustrate the problem:

**Example 1:** Assume $X = Y = \mathbb{R}$, let $x$ have a gaussian prior with mean 0 and unit variance and let $y$ be gaussian with mean $x$ and unit variance. In this case we have
$$
\mathbb{E}(x | y)
=
\frac{y}{2}
$$
which is continuous.

**Example 2:** Assume $X = Y = \ell_2$. Let $x$ have a gaussian prior with $x_i \sim N(0, 1/i)$ and let $y_i ~ \sim N(x_i / i, 1 / i^3)$. In this case we have
$$
[\mathbb{E}(x | y)]_i =
\left(
\frac{1}{i}
i^3
\frac{1}{i} +
i
\right)^{-1}
\frac{1}{i}
i^3
y_i
=
\frac{i}{2}
y_i
$$
This is a linear operator, but unbounded, hence $\mathbb{E}(x | \cdot)$ is not continuous.