Convergence in inner product spaces I’m told a Hilbert space is a complete inner product space and so any Hilbert space is also a Banach space. 
If we have a inner product space $(H,\langle \cdot, \cdot \rangle )$. What does it mean for a sequence $(x_n)$ in $H$ to converge? 
And what does it mean to be complete in the sense of an inner product space? 
Does it just mean that the sequence converges with respect to the norm induced by the inner product?

The inner product gives rise to a metric $d$ defined by $d(x,y)=\sqrt { \langle x-y, x-y \rangle}$. Convergence and completeness are defined using this metric or w.r.t. the norm defined by $\|x-y\|= \sqrt { \langle x-y, x-y \rangle}$.
