If ##A \subset B## are linear subspaces of a Hilbert space, ##B \ominus A = \{x \in B: (x,y) = 0 \text{ for all }y \in A\}##. ##\ominus## is also used for the symmetric difference of sets.
The symbol ##\ominus## represents the orthogonal complement in Hilbert Spaces. It is used to denote the set of all vectors that are perpendicular to a given vector or subspace in a Hilbert Space.
In Hilbert Spaces, the orthogonal complement ##\ominus## of a subspace ##U## is defined as the set of all vectors ##v## such that ##\langle v, u \rangle = 0## for all ##u## in ##U##. In other words, the inner product of any vector in the orthogonal complement with any vector in the subspace is equal to zero.
Consider a Hilbert Space ##\mathbb{R}^3## with the standard inner product. Let ##U## be the subspace spanned by the vectors ##(1,0,0)## and ##(0,1,0)##. The orthogonal complement ##U^\perp## of ##U## is the set of all vectors ##(x,y,z)## such that ##x=0## and ##y=0##, which can be represented as ##U^\perp = \{(0,0,z) | z \in \mathbb{R}\}##.
The orthogonal complement ##\ominus## is an important concept in Hilbert Spaces as it allows us to decompose a space into two mutually orthogonal subspaces. This decomposition is useful in many areas of mathematics, including functional analysis and linear algebra.
The concept of ##\ominus## is closely related to the concept of orthogonality in Hilbert Spaces. Two vectors are orthogonal if their inner product is equal to zero. Similarly, a vector and a subspace are orthogonal if their inner product is equal to zero for all vectors in the subspace. This is the same condition that defines the orthogonal complement ##\ominus## of a subspace in Hilbert Spaces.