Normed Space and Preliminaries
Before proceeding, the reader is advised to be familiar with a few foundational concepts in mathematical analysis. In particular, a basic understanding of metric spaces and vector spaces is essential, as they provide the structural framework in which our discussion takes place. Additionally, the reader should be comfortable with the definition of continuity and limit, as well as the notions of differentiation and integration in the classical sense. These topics form the core analytical tools we will rely on throughout this work.
Definition and examples of Normed Spaces
Let be a vector space over , A norm on is a function such tat for all and ,
- and if and only if
- =
- .
A vector space with norm is called a normed vector space or simply normed space.
Let the space vectorial . The function defined by
is a norm in . If is or just then .
Vector Space of Continuous Functions and their norm
Let the set of all functions that are continuos on . A function if and only if,
in other words, is continuos at every point
Let be the vector espaces of continuos functions. The function defined by
is a norm in .
Vector Space of Lebesgue Functions and their norm
Let be the vector space of measurable functions The function defined by
is a norm on
Sequence
The notion of a sequence is fundamental in analysis. In fact, entire books could be written based solely on sequences. Many core definitions such as continuity, differentiation, integration, convergence, density, compactness, completeness, closure, etc, can be formulated or understood using sequences.
Let be a normed spaces. A sequence in is an ordered list of elements of , written in the form , typically denoted by . Formally, it is a function such that for each .
A sequence in a normed space is convergent (and converges to ), if, for every , there exists such that for all As usual, we write
A subset of a normed spaces is said to be dense in , if, for each is the limit of a sequence of elements of .
Cauchy Sequence and Completeness
We now introduce the notion of Cauchy sequences, which is essential for understanding convergence in normed spaces. In particular, this concept allows us to formally define Banach spaces.
A sequence in a normed space is a Cauchy sequence if, for every , there exists such that for all
Although Banach spaces will not be used extensively in this research, we consider it appropriate to include them for the sake of completeness and to provide a solid foundational framework
A Banach space is a normed space in which every Cauchy sequence is convergent. This property is also known as completeness of the space.
Bounded Linear Transformation
We now briefly introduce the definition of a bounded linear transformation, as it will play a fundamental role in the development of this research. For this, we will quickly recall the definition of linear transformation.
Let be vector spaces over . A function is called a linear transformation if, for all and ,
Let and be normed spaces and let be a linear transformation. is said to be bounded if there exists a positive real number such that for all
The norm of T is defined by
Let be a normed space over . The set of all bounded linear transformation from to is called the dual space of and is denoted by
References
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B. P. Rynne and M. A. Youngson, Linear Functional Analysis, 2nd ed. London: Springer, 2008, pp. 12, 31-33.
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R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd ed. Amsterdam: Academic Press, 2003, pp. 5.