Connections between metrics, norms and scalar products (for understanding e.g. Banach and Hilbert spaces) I am trying to understand the differences between 
$$
\begin{array}{|l|l|l|}
\textbf{vector space} & \textbf{general} & \textbf{+ completeness}\\\hline
\text{metric}& \text{metric space} & \text{complete space}\\
\text{norm} & \text{normed} & \text{Banach space}\\
\text{scalar product} & \text{pre-Hilbert space}  & \text{Hilbert space}\\\hline
\end{array}
$$
What I don't understand are the differences and connections between metric, norm and scalar product. Obviously, there is some kind of hierarchy but I don't get the full picture.
Can anybody help with some good explanations/examples and/or readable references?

When is a normed subspace of vector space a pre-Hilbert space.  The analogous concept to orthogonal and orthonormal sequences in a normed space is perpendicular and a perpnormal sequences.  If the norm squared of the sum of a linear combination of non-zero vectors equals the sum of the norm squared of each of the components, then the set of vectors is perpendicular.  For example, x,y are perpendicular vectors in a complex normed space, if for arbitrary complex numbers a,b,
||ax + by||2 = ||ax||2 + ||by||2
= |a|2 ||x||2 + |b|2 ||y||2,
and perpnormal, if, ||x||2 = ||y||2 = 1, so,
||ax + by||2 = |a|2 + |b|2.
Define the polarization product of two vectors x,y in a normed space using the polarization identity in a pre-Hilbert space,
(x|y) = 1/4 { ||x + y||2 - ||x - y||2 + i ||x + i y||2 - i ||x – i y||2 }.
Then, a normed space having a sequence of perpnormal vectors (vectors that are perpendicular and unit vectors), is equivalent to all pairs of vectors in the normed space satisfying the parallelogram law. learn more>>
