# Orthogonal and Symplectic Separation

• Herbert Gross
Part of the Progress in Mathematics book series (PM, volume 1)

## Abstract

When one has to handle two mutually perpendicular subspaces F and G in a sesquilinear space (E, Φ) it is often a great advantage if E splits orthogonally such that F and G are contained in summands,
$$E = {E_1}\mathop \oplus \limits^ {E_2},F \subset {E_1},G \subset {E_2}$$
(0)
If this happens then we say that F and G can be orthogonally separated in E. From (0) we read off that
$${(F + G)^{ \bot \bot }} = {F^{ \bot \bot }} + {G^{ \bot \bot }}$$
(1)
$${F^ \bot } + {G^ \bot } = E$$
(2)
.

## Keywords

Lattice Versus Modular Lattice Isotropic Subspace Finite Dimensional Subspace Isotropic Vector
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## References to Chapter VI

1. [1]
L. Brand, Erweiterung von algebraischen Isometrien in sesquilinearen Räumen. PhD Thesis, Univ. of Zurich 1974.Google Scholar
2. [2]
F. K. Fischer, Orthogonale und symplektische Zerlegung sesquilinearer Räume. Master’s Thesis, Univ. of Zurich 1977.Google Scholar
3. [3]
A. Frapolli, Generalizzazione di un teorema di H.A. Keller sulla modularità del reticolo dei sottospazi ortogonalmente chiusi di uno spazio sesquilineare. Master’s Thesis, Univ. of Zurich 1975. (This concerns some technicalities when char k = 2; in [6] it is assumed that char k ≠ 2.)Google Scholar
4. [4]
H. Gross and P. Hafner, The sublattice of an orthogonal pair in a modular lattice. Ann. Acad. Sci. Fenn. vol. 4 1978/1979.Google Scholar
5. [5]
B. Jónsson, Distributive sublattices of a modular lattice. Proc. Amer. Math. Soc. 6 (1955) 682–688.
6. [6]
H. A. Keller, Ueber den Verband der orthogonal abgeschlossenen Teilräume eines hermiteschen Raumes. Letter to the author of Nov. 7, 1973, pp. 1–6.Google Scholar