In this article we will introduce the definition of Reproducing Kernel Hilbert Space (RKHS).

In revisiting the proofs of many theorems in mathematics, we often need to use sequential compactness. However, the Riesz lemma in normed linear spaces tells us that every bounded set in an infinite-dimensional normed linear space is not sequentially compact, which brings many inconveniences. In fact, the essential reason for this phenomenon is that the topology induced by the norm in a normed linear space is “too strong.” For this reason, we need to introduce the weak topology, so that bounded sets are sequentially compact in the weak-topology sense. In this article, we mainly review local convex topological vector spaces and some main results on weak convergence.