# On Some Results on Linear Orthogonality Spaces

@article{Kanu2014OnSR, title={On Some Results on Linear Orthogonality Spaces}, author={R. U. Kanu and K. Rauf}, journal={Asian Journal of Mathematics and Applications}, year={2014}, volume={2014} }

The existence of an inner product provides a means of introducing the notion of orthogonality in normed linear spaces. The most natural definition of orthogonality is: “x⊥y, if and only if the inner product is zero.” This paper introduces different notions of orthogonality in normed linear spaces and shows that they are all equivalent in real Hilbert spaces. Also, some characterizations of linear orthogonality spaces are given.

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#### References

SHOWING 1-10 OF 26 REFERENCES

An extension of the notion of orthogonality to Banach spaces

- Mathematics
- 2002

Abstract We extend the usual notion of orthogonality to Banach spaces. We show that the extension is quite rich in structure by establishing some of its main properties and consequences. Geometric… Expand

Characterisation of orthogonality in certain Banach spaces

- Mathematics
- Bulletin of the Australian Mathematical Society
- 2002

In this paper we adopt the notion of orthogonality introduced by the author in a previous article. We establish a characterisation for orthogonality in the spaces , where S is a set of positive… Expand

On orthogonally exponential and orthogonally additive mappings

- Mathematics
- 1997

Let E be a real inner product space, (F,+) an abelian σ-bounded topological group, and K a discrete subgroup of F . It is proved that (under suitable assumptions on E) the Christensen and Baire… Expand

Introductory Functional Analysis With Applications

- Mathematics
- 1978

Metric Spaces. Normed Spaces Banach Spaces. Inner Product Spaces Hilbert Spaces. Fundamental Theorems for Normed and Banach Spaces. Further Applications: Banach Fixed Point Theorem. Spectral Theory… Expand

Orthogonality and linear functionals in normed linear spaces

- Mathematics
- 1947

The natural definition of orthogonality of elements of an abstract Euclidean space is that x ly if and only if the inner product (x, y) is zero. Two definitions have been given [11](2) which are… Expand

Two mappings related to semi-inner products and their applications in geometry of normed linear spaces

- Mathematics
- 2000

In this paper we introduce two mappings associated with the lower and upper semi-inner product (·, ·)i and (·, ·)S and with semi-inner products [·, ·] (in the sense of Lumer) which generate the norm… Expand

Orthogonally additive and orthogonally increasing functions on vector spaces.

- Mathematics
- 1975

A real-valued function /: X—> R on an inner product space X is orthogonally additive if f(x + y) = /(JC) + f(y) whenever x l y . We extend this concept to more general spaces called orthogonality… Expand

Explicit characterizations of orthogonality in lSp(C)

- Mathematics
- 2002

Abstract In a previous paper, the author used a notion of orthogonality introduced in another article to establish characterizations for orthogonality in the spaces l S p ( C ), 1⩽ p l S 2 ( C ), via… Expand

SEMI-INNER-PRODUCT SPACES

- Mathematics
- 1961

function as a particular Banach space (whose norm satisfies the parallelogram law), but rather as an inner-product space. It is in terms of the innerproduct space structure that most of the… Expand

Inner products in normed linear spaces

- Mathematics
- 1947

Let T be any normed linear space [l, p. S3]. Then an inner product is defined in T if to each pair of elements x and y there is associated a real number (x, y) in such a way that (#, y) » (y, x),… Expand