Parallelogram Law - Normed Vector Spaces Satisfying The Parallelogram Law

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:

\begin{align} \langle x, y\rangle&={\|x+y\|^2-\|x-y\|^2\over 4}\\ &=\frac{1}{4} \left\\ &=\frac{1}{4} \left\\ &=(x\cdot y), \end{align}

which is the standard dot product of two vectors.

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