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.