Orthogonality

A generalization of the concept of perpendicularity of vectors in a Euclidean space. The most natural concept of orthogonality is put forward in the theory of Hilbert spaces. Two elements $x$ and $y$ of a Hilbert space $H$ are said to be orthogonal $( x \perp y)$ if their inner product is equal to zero ( $( x, y) = 0$). This concept of orthogonality in the particular case where $H$ is a Euclidean space coincides with the concept of perpendicularity of two vectors. In terms of this concept, in any Hilbert space Pythagoras' theorem holds: If an element $x \in H$ is equal to a finite or countable sum of pairwise orthogonal elements $x _ {i} \in H$( the countable sum $\sum _ {i=} 1 ^ \infty x _ {i}$ is understood in the sense of convergence of the series in the metric of $H$), then $\| x \| ^ {2} = \sum _ {i=} 1 ^ \infty \| x _ {i} \| ^ {2}$( see Parseval equality).

A complete, countable, orthonormal system $\{ x _ {i} \}$ in a separable Hilbert space is the analogue of a complete system of pairwise orthonormal vectors in a finite-dimensional Euclidean space: Any element $x \in H$ can be uniquely represented as the sum $\sum _ {i=} 1 ^ \infty c _ {i} x _ {i}$, where $c _ {i} x _ {i} = ( x, x _ {i} ) x _ {i}$ is the orthogonal projection of the element $x$ onto the span of the vector $x _ {i}$.

E.g., in the function space $L _ {2} [ a, b]$, if $\{ \phi _ {k} \}$ is a complete orthonormal system, then for every $f \in L _ {2} [ a, b]$,

$$f = \sum _ { k= } 1 ^ \infty c _ {k} \phi _ {k}$$

in the metric of the space $L _ {2} [ a, b]$, where

$$c _ {k} = \int\limits _ { a } ^ { b } f ( x) \overline{ {\phi _ {k} ( x) }}\; dx.$$

When the $\phi _ {k}$ are bounded functions, the coefficients $c _ {k}$ can be defined as above for any integrable function. In these cases the question of the convergence of a corresponding series in one sense or another is of interest (see Trigonometric system; Haar system). With respect to functions, therefore, the term "orthogonality" is used in a broader sense: Two functions $f$ and $g$ which are integrable on the segment $[ a, b]$ are orthogonal if

$$\int\limits _ { a } ^ { b } f( x) g( x) dx = 0$$

(for the integral to exist, it is usually required that $f \in L _ {p} [ a, b]$, $1 \leq p \leq \infty$, $g \in L _ {q} [ a, b]$, $p ^ {- 1 } + q ^ {- 1 } = 1$, where $L _ \infty [ a, b]$ is the set of bounded measurable functions).

Definitions of orthogonality of elements of an arbitrary normed linear space also exist. One of them (see [4]) is as follows: An element $x$ of a real normed space $B$ is considered orthogonal to the element $y$ if $\| x \| \leq \| x + ky \|$ for all real $k$. In terms of this concept certain necessary and sufficient conditions have been established under which a scalar (inner) product of elements of $B$ can be defined (see [5], [6]).

References

 [1] L.V. Kantorovich, G.P. Akilov, "Functionalanalysis in normierten Räumen" , Akademie Verlag (1964) (Translated from Russian) [2] N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Wiley, reprint (1988) [3] S. Kaczmarz, H. Steinhaus, "Theorie der Orthogonalreihen" , Chelsea, reprint (1951) [4] G. Birkhoff, "Orthogonality in linear metric spaces" Duke Math. J. , 1 (1935) pp. 169–172 [5] R. James, "Orthogonality and linear functionals in normed linear spaces" Trans. Amer. Math. Soc. , 61 (1947) pp. 265–292 [6] R. James, "Inner products in normed linear spaces" Bull. Amer. Math. Soc. , 53 (1947) pp. 559–566