Normed Vector Spaces Satisfying The Parallelogram Law
Most real and complex normed vector spaces do not have inner products, but all normed vector spaces have norms (by definition). For example, a commonly used norm is the p-norm:
where the are the components of vector .
Given a norm, one can evaluate both sides of the parallelogram law above. A remarkable fact is that if the parallelogram law holds, then the norm must arise in the usual way from some inner product. In particular, it holds for the p-norm if and only if p = 2, the so-called Euclidean norm or standard norm.
For any norm satisfying the parallelogram law (which necessarily is an inner product norm), the inner product generating the norm is unique as a consequence of the polarization identity. In the real case, the polarization identity is given by:
or, equivalently, by:
In the complex case it is given by:
For example, using the p-norm with p = 2 and real vectors, the evaluation of the inner product proceeds as follows:
which is the standard dot product of two vectors.
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