An orthogonal sum 
is a direct sum 
of two subspaces
of a vector space 
, equipped with a symmetric
bilinear form, an alternating bilinear
form, or a Hermitian form 
, such that 
for all 
and 
(Conrad, pp. 6-7). In particular, the direct
sum condition requires 
. For a positive integer 
, the notation
 |
denotes a direct sum of subspaces of 
which are pairwise orthogonal.
For vector spaces 
and 
over the same field, with symmetric
bilinear forms 
and 
, their orthogonal sum is the vector
space 
with the symmetric bilinear form
 |
The same construction applies to two alternating bilinear forms (Elman et al. 2008, p. 14). The subspaces
consisting of vectors of the forms 
and 
are mutually orthogonal
(Conrad, pp. 6-7).
Even when two subspaces are mutually orthogonal, their sum need not be a direct
sum. For example, in 
with the symmetric
bilinear form 
,
the subspace 
is orthogonal to itself.
Taking 
gives 
, so their sum
inside 
is not a direct sum (Conrad, p. 6).
