We will begin by providing a brief introduction to self-normalizing concentration inequalities for scalar and finite-dimensional martingales, which are very useful in measuring the size of a stochastic process in terms of another, growing, process (the 2009 book of de la Pena et al is a good reference). We will then present a self-normalizing concentration inequality for martingales that live in the (potentially infinite-dimensional) Reproducing Kernel Hilbert Space (RKHS) of a p.s.d. kernel. We will conclude by illustrating applications to online kernel least-squares regression and multi-armed bandits with infinite action spaces, a.k.a. sequential noisy function optimization [Joint work with Sayak Ray Chowdhury (IISc)].