THE f-NORM; A GENERALIZATION OF THE NORM OF FUNCTIONAL ANALYSIS

A generalisation, f-norm, of the concept of norm as defined in Functional Analysis is introduced. The norm of x  is  times the norm of x . In the definiton of the f-norm, this requirement is relaxed, so that the f-norm of x  is ) ( f times the f-norm of x , where f is a function satisfying certain properties, which are presented. Examples of f-norms are given, the metric induced by an f-norm is considered and results concerning continuity of certain functions obtained. Sets that are convex with respect to an f-norm are studied. Limits with respect to the f-norm are also considered. Indexing terms/


INTRODUCTION
The concept of norm is fundamental in Mathematics and various types of norm are used according to the requirements in different areas. It is well known that norms have been used extensively in the study of functions. New types of norm continue to be introduced periodically. Urbanski [7] gave a generalization of the Koshi-Shimogaki and Amemiya's norm defined in a modular space over a field with valuation. Lee and Verleysen [2] considered a generalization of the p L norm suitable for time-series data with particular regard for the temporal structure. Ramsey and Silverman [4] and Roni, Delannay, Conan-Guez and Verleysen [5] also took the temporal structure of data into account. Hatami [1] studied generalizations of Gower's norm, which are defined in terms of integrals. He obtained conditions on the structure of integrals which corresponds to norm functions. Matkowski [3] gave some results about a functional which becomes the norm  p L under certain conditions. One of the conditions that a norm must satisfy is that the norm of x  be equal to x  . In this paper, we introduce a generalization of the concept of norm, namely, the f-norm, which eliminates the strict requirement that the norm of x  be equal to the modulus of lambda times the norm of x. We require instead that the norm of x  be equal to a function f of lambda times the norm of x. We present some properties of this function, provide examples of f-norms, consider the metric introduced by an f-norm and obtain results concerning continuity of certain functions. Many of the results presented in this paper are generalizations of known results about norms. In some cases, the proofs in this paper are quite similar to proofs of corresponding results about norms, but in other cases, the proofs presented here are significantly different. Basically, our purpose is to consider how results that are given with respect to the usual norm may be modified when the norm is replaced by the f-norm.

Definitions and Notations
Notation 2.1. ₵ represents the set of complex numbers and  represents the set of real numbers.
2. An f-norm on a real or complex linear space  is a non-negative real-valued function  on  with the following properties: where f is a continuous, real-valued monotone-increasing function on the non-negative real numbers such that the gradient of f is positive and non-increasing.
,and the gradient of the graph of Hence the gradient of this graph >1 for all point of [0,1], and so (1,1) cannot be on the graph. However, we know that (1,1) is on the graph and hence

. The metric induced by the f-norm
The following corollary follows immediately. The following theorem is a generalization of a well-known result relating metrics to norms, (cf Simmons [6] page 54) Theorem 4.3.   is a metric on  .
Proof.   satisfies the conditions to be a metric as shown below.
Hence g is continuous on .
 then the  -valued mapping given by Hence g is continuous.

Examples of f-norms
Example 5.1. The f-norm described here is similar to the norm on p L spaces (c.f. Simmons [6], page 215). Let ) (S L p be the set of real-valued, continuous functions g defined on an interval S of the real-numbers  where f satisfies the conditions: f is a continuous, real-valued function on the non-negative real numbers (iii) the gradient of f is non-increasing Let g be a continuous, real-valued function defined on the real numbers, and let the norm of g be given The following theorem provides with a way to obtain f-norms by choosing the function f in a suitable manner. (c.f. page 215 Simmons [6]).