An inner product naturally induces an associated norm, (denoted |x| and |y| in the picture); so, every inner product space is a normed vector space. If this normed space is also complete (that is, a Banach space) then the inner product space is a Hilbert space. If an inner product space is not a Hilbert space, it can be extended by completion to a Hilbert space \overline{H}. This means that H is a linear subspace of \overline{H}, the inner product of H is the restriction of that of \overline{H}, and H is dense in \overline{H} for the topology defined by the norm.
