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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

Definition of note normed space

Let XX be a vector space over R\mathbb{R}, A norm on XX is a function :XR\lVert \cdot \rVert: X\to\mathbb{R} such tat for all x,yXx,y \in X and αR\alpha \in\mathbb{R},

  1. x0\lVert x\rVert\geq 0 and x=0\lVert x\rVert = 0 if and only if x=0,x=0,
  2. αx\lVert\alpha x\rVert = αx,|\alpha|\lVert x\rVert,
  3. x+yx+y\lVert x + y \rVert \leq \lVert x \rVert + \lVert y \rVert.

A vector space XX with norm is called a normed vector space or simply normed space.

Example

Let the space vectorial Rn\mathbb{R}^n. The function :RnR\lVert \cdot \rVert: \mathbb{R}^n\to\mathbb{R} defined by

x=(x1,...,xn)=j=1nxj2\lVert x\rVert = \lVert (x_1, ..., x_n)\rVert = \sqrt{\sum^n_{j=1}|x_j|^2 }

is a norm in Rn\R^n. If is R1\mathbb{R}^1 or just R\mathbb{R} then x=x\lVert x\rVert=|x|.

Vector Space of Continuous Functions and their norm

Let C(Ω)C(\Omega) the set of all functions f:ΩRnRf:\Omega\sub\mathbb{R}^n\to\mathbb{R} that are continuos on Ω\Omega. A function fC(Ω)f\in C(\Omega) if and only if,

ε>0, δ>0 such that if xa<δ, then f(x)f(a)<ε\forall \varepsilon > 0,\ \exists \delta > 0 \ \text{such that if } \|x - a\| < \delta, \text{ then } |f(x) - f(a)| < \varepsilon

in other words, ff is continuos at every point aΩ.a \in\Omega.

Example

Let C(Ω)C(\Omega) be the vector espaces of continuos functions. The function :C(Ω)R\lVert \cdot \rVert _\infty: C(\Omega)\to\mathbb{R} defined by

f=sup{f(x):xΩ}\lVert f \rVert _\infty = \sup \{|f(x)|: x\in\Omega\}

is a norm in C(Ω)C(\Omega).

Vector Space of Lebesgue Functions and their norm

Example

Let Lp(Ω)L^p(\Omega) be the vector space of measurable functions f:ΩR.f : \Omega \to \mathbb{R}. The function p:Lp(Ω)R\lVert \cdot \rVert _p : L^p(\Omega) \to \mathbb{R} defined by

fp=(Ωf(x)pdμ(x))1/p\lVert f \rVert _p = \left( \int_\Omega |f(x)|^p \, d\mu(x) \right)^{1/p}

is a norm on Lp(Ω).L^p(\Omega).

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.

Definition of sequence

Let XX be a normed spaces. A sequence in XX is an ordered list of elements of XX, written in the form (x1,x2,...)(x_1, x_2, ...), typically denoted by {xn} \{x_n\}. Formally, it is a function s:NXs:\mathbb{N}\to X such that xn=s(n)x_n=s(n) for each nNn\in\mathbb{N}.

A sequence {xn}\{x_n\} in a normed space XX is convergent (and converges to xx), if, for every ϵ>0\epsilon>0, there exists NNN\in\mathbb{N} such that xxn<ϵ\lVert x - x_n \rVert<\epsilon for all nN.n\ge N. As usual, we write limnxn=x.\lim_{n\to\infty} x_n=x.

A subset AA of a normed spaces XX is said to be dense in XX, if, for each xXx\in X is the limit of a sequence of elements of AA.

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.

Definition of Cauchy Sequence

A sequence {xn}\{x_n\} in a normed space XX is a Cauchy sequence if, for every ϵ>0\epsilon > 0, there exists NNN \in \mathbb{N} such that xmxn<ϵ\lVert x_m - x_n \rVert < \epsilon for all n,mN.n, m \ge N.

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

Definitions of Banach Space and Completeness

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 V,WV, W be vector spaces over R\mathbb{R}. A function T:VWT: V \to W is called a linear transformation if, for all α,βR\alpha, \beta \in \mathbb{R} and x,yVx, y \in V,

T(αx+βy)=αT(x)+βT(y).T(\alpha x + \beta y) = \alpha T(x) + \beta T(y).
Definition of bounded linear transformation

Let XX and YY be normed spaces and let T:XYT:X\to Y be a linear transformation. TT is said to be bounded if there exists a positive real number kk such that T(x)kx\lVert T(x)\rVert \leq k\lVert x\rVert for all xX.x \in X.

The norm of T is defined by T=sup{T(x):x1}.\lVert T\rVert = \sup\{\lVert T(x) \rVert: \lVert x\rVert \leq 1 \}.

Definition of Dual Space

Let XX be a normed space over R\mathbb{R}. The set of all bounded linear transformation from XX to R\mathbb{R} is called the dual space of XX and is denoted by X.X'.

References

  1. B. P. Rynne and M. A. Youngson, Linear Functional Analysis, 2nd ed. London: Springer, 2008, pp. 12, 31-33.

  2. R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd ed. Amsterdam: Academic Press, 2003, pp. 5.