What is actually weak convergence

I saw different definitions from two sources. I am getting confused
what it actually is.
For example, let (xn) is a sequence in H and let  y and x  belong to
H.  xn is convergent weakly to x iff <xn,y>-><x,y> for all y belonging
to H.

but in some other texts, they mentioned about dual space. Please let
me know what is actually is. Is the definition depending upon the
space
we are talking about?