  We study algorithms for online linear optimization in Hilbert spaces,
focusing on the case where the player is unconstrained. We develop a novel
characterization of a large class of minimax algorithms, recovering, and even
improving, several previous results as immediate corollaries. Moreover, using
our tools, we develop an algorithm that provides a regret bound of
$\mathcal{O}\Big(U \sqrt{T \log(U \sqrt{T} \log^2 T +1)}\Big)$, where $U$ is
the $L_2$ norm of an arbitrary comparator and both $T$ and $U$ are unknown to
the player. This bound is optimal up to $\sqrt{\log \log T}$ terms. When $T$ is
known, we derive an algorithm with an optimal regret bound (up to constant
factors). For both the known and unknown $T$ case, a Normal approximation to
the conditional value of the game proves to be the key analysis tool.
