  Following [3] we say that a Tychonoff space $X$ is an Ascoli space if every
compact subset $\mathcal{K}$ of $C_k(X)$ is evenly continuous; this notion is
closely related to the classical Ascoli theorem. Every $k_\mathbb{R}$-space,
hence any $k$-space, is Ascoli.
  Let $X$ be a metrizable space. We prove that the space $C_{k}(X)$ is Ascoli
iff $C_{k}(X)$ is a $k_\mathbb{R}$-space iff $X$ is locally compact. Moreover,
$C_{k}(X)$ endowed with the weak topology is Ascoli iff $X$ is countable and
discrete.
  Using some basic concepts from probability theory and measure-theoretic
properties of $\ell_1$, we show that the following assertions are equivalent
for a Banach space $E$: (i) $E$ does not contain isomorphic copy of $\ell_1$,
(ii) every real-valued sequentially continuous map on the unit ball $B_{w}$
with the weak topology is continuous, (iii) $B_{w}$ is a $k_\mathbb{R}$-space,
(iv) $B_{w}$ is an Ascoli space.
  We prove also that a Fr\'{e}chet lcs $F$ does not contain isomorphic copy of
$\ell_1$ iff each closed and convex bounded subset of $F$ is Ascoli in the weak
topology. However we show that a Banach space $E$ in the weak topology is
Ascoli iff $E$ is finite-dimensional. We supplement the last result by showing
that a Fr\'{e}chet lcs $F$ which is a quojection is Ascoli in the weak topology
iff either $F$ is finite dimensional or $F$ is isomorphic to the product
$\mathbb{K}^{\mathbb{N}}$, where $\mathbb{K}\in\{\mathbb{R},\mathbb{C}\}$.
