Springer International Publishing AG,
part of Springer Nature 2018
On mappings which approximately preserve angles
S. M. S. Nabavi Sales
Abstract. In this paper, we study mappings, which approximately preserve angles between
inner product spaces. We also introduce a notion of angle in normed spaces. The notion
of angle, considered in this part, relates to the well-known Birkhoﬀ–James orthogonality.
Based on it, we express a characterization for approximate Birkhoﬀ–James orthogonality,
introduced in the literature, through this notion of angle. Then we return to the issue of
mappings which approximately preserve angle stating some results in normed spaces.
Mathematics Subject Classiﬁcation. 47B49, 46C05, 46B20.
Keywords. Angle in normed spaces, Birkhoﬀ–James orthogonality, Approximate orthogonal-
ity preserving, Approximate angle preserving.
In this paper, we deal with mappings which approximately preserve angles.
This issue arises, originally, from investigations on complex functions. A com-
plex function that preserves angles and distances against any extreme defor-
mation, is likely to preserve many geometric aspects, approximately. In a real
inner product space (X , ., .), the angle between two non-zero vectors x and
y is deﬁned, routinely, by invoking the inner product. Indeed x∠
y for some
θ ∈ [0,π] if and only if cos θ =
. We also say that x∠
y, for some θ ∈ [0,π]
and some ε, whenever |
− cos θ|≤ε. In general normed spaces in which
norm does not necessarily come from an inner product, various types of deﬁni-
tions were oﬀered. For instance, some investigations in this ﬁeld carried out in
[12,13], during the 80s in which the authors introduced some types of orthog-
onality and some angle concept in normed spaces. They, among other things,
introduce a notion of angle A(x, y) between two vectors x and y in a normed
space X . The deﬁnition of angle provides the means of deﬁning orthogonality
between vectors; see also .