For a vector in a normed space , define the orthogonal complement to be the set of all vectors such that for all scalars . In an inner product space (real or complex), this agrees with the normal definition of orthogonality because as , and the right hand side can be nonnegative only if .
Let’s see what properties of orthogonal complement survive in a general normed space. For one thing, if and only if . Another trivial property is that for all . More importantly, is a closed set that contains some nonzero vectors.
- Closed because the complement is open: if for some , the same will be true for vectors close to .
- Contains a nonzero vector because the Hahn-Banach theorem provides a norming functional for , i.e., a unit-norm linear functional such that . Any is orthogonal to , because .
In general, is not a linear subspace; it need not even have empty interior. For example, consider the orthogonal complement of the first basis vector in the plane with (taxicab) metric: it is .
This example also shows that orthogonality is not symmetric in general normed spaces: but . This is why I avoid using notation here.
In fact, is the union of kernels of all norming functionals of , so it is only a linear subspace when the norming functional is unique. Containment in one direction was already proved. Conversely, suppose and define a linear functional on the span of so that . By construction, has norm 1. Its Hahn-Banach extension is a norming functional for that vanishes on .
Consider as an example. A function satisfies precisely when its th moment is minimal among all translates . This means, by definition, that its “-estimator” is zero. In the special cases the estimator is known as the median, mean, and midrange, respectively. Increasing gives more influence to outliers, so is the more useful range for it.