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

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

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

Example

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

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

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

Vector Space of Continuous Functions and their norm

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

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

in other words, $f$ is continuos at every point $a \in\Omega.$

Example

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

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

is a norm in $C(\Omega)$.

Vector Space of Lebesgue Functions and their norm

Example

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

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

is a norm on $L^p(\Omega).$

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

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

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

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 $\{x_n\}$ in a normed space $X$ is a Cauchy sequence if, for every $\epsilon > 0$, there exists $N \in \mathbb{N}$ such that $\lVert x_m - x_n \rVert < \epsilon$ for all $n, m \ge N.$

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.

Example

$L^p(\Omega)$ is a Banach 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 $V, W$ be vector spaces over $\mathbb{R}$. A function $T: V \to W$ is called a linear transformation if, for all $\alpha, \beta \in \mathbb{R}$ and $x, y \in V$,

$$T(\alpha x + \beta y) = \alpha T(x) + \beta T(y).$$

Example

Let $\mathcal{P}$ be the vectorial subspace of $C[a,b]$ consisting of all polynomical functions p.

$T: \mathcal{P}\to \mathcal{P}$ is the linear transformation define by

$$T(p)=Dp.$$

Definition of bounded linear transformation

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

The set of all bounded linear transformation from $X$ to $Y$ is denoted by $B(X,Y).$

Example

The linear transformation given in the previous example is unbounded with the norm of the supreme $\lVert \cdot \rVert _\infty$.

Example

Let $T: \mathbb{R}^2\to \mathbb{R}$ the linear transformation define by

$$T(x)=T(x_1,x_2)=x_1+x_2$$

is bounded with the norm $\mathbb{R}^2$,

$$\lVert T(x) \rVert = |x_1+x_2|\leq |x_1|+|x_2|\leq \sqrt{2}\sqrt{x^2_1+x^2_2}=\sqrt{2}\lVert x \rVert.$$

Example

Let $T: C([a,b])\to \mathbb{R}$ the linear transformation define by

$$T(f)=\int_a^bf(t)dt$$

is bounded on $C([a,b])$ with the norm of the supreme,

$$\lVert T(f) \rVert = | T(f) | \leq (b-a)\lVert f \rVert _\infty.$$

Example

Let $T_{x_0}: C([a,b])\to \mathbb{R}$ the linear transformation define by

$$T_{x_0}(f)=f(x_0)$$

is bounded on $C([a,b])$ with the norm of the supreme,

$$\lVert T_{x_0}(f) \rVert = |f(x_0)| \leq \lVert f \rVert _\infty.$$

Dual Space

Example

Let $X$ and $Y$ be normed spaces and let $T:X\to Y$ be a bounded linear transformation. The function $\lVert \cdot \rVert: B(X,Y)\to \mathbb{R}$ defined by

$$\lVert T \rVert = \sup \{\lVert T(x) \rVert : \lVert x \rVert\leq 1\}$$

is a norm on $B(X,Y)$

warning

$\lVert T \rVert$ is not the same as $\lVert T(x) \rVert$.

Definition of Dual Space

Let $X$ be a normed space over $\mathbb{R}$. The dual space of $X$, denoted by $X'$, is the set of all bounded linear transfomation

$$T : X \to \mathbb{R}.$$

Or simply $B(X,\mathbb{R}).$ Equipped with the operator norm $\lVert T \rVert$ the dual space $X'$ is itself a normed space.

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.