Infinite Dimensional Linking

Abstract

Let N be a closed, separable subspace of a Hilbert space E. We can define a new norm |v|_w satisfying |v|_w ≤∥v∥, ∀v ∈ N and such that the topology induced by this norm is equivalent to the weak topology of N on bounded subsets of N. This can be done as follows: Let {e _k} be an orthonormal basis for N. Define

$$\displaystyle (u,v)_w=\sum _{k=1}^{\infty }\frac {(u,e_k)(v, e_k)}{2^k}, \quad u,v\in N. $$