1.1 Definition and properties of normed linear spaces

Normed linear spaces are vector spaces with a special function that measures vector size. This function, called a norm, must follow specific rules like being non-negative and obeying the triangle inequality.

These spaces are crucial in functional analysis. They allow us to measure distances between vectors, study convergence of sequences, and analyze continuity of functions in abstract settings beyond just regular Euclidean space.

Normed Linear Spaces

Definition of normed linear spaces

• Vector space $V$ over a scalar field $\mathbb{F}$ ($\mathbb{R}$ or $\mathbb{C}$) equipped with a norm function $\|\cdot\|: V \to \mathbb{R}$
  • Assigns a non-negative real number to each vector in the space, measuring its "length" or "size"
  • Induces a metric $d$ on the space, defined by $d(x, y) = \|x - y\|$ for all $x, y \in V$, measuring the "distance" between vectors
• Must satisfy the following properties for all $x, y \in V$ and $\alpha \in \mathbb{F}$:
  • Positive definiteness: $\|x\| \geq 0$ and $\|x\| = 0$ if and only if $x = 0$, ensuring zero vector has zero norm and non-zero vectors have positive norms
  • Homogeneity: $\|\alpha x\| = |\alpha| \|x\|$, scaling a vector by a scalar scales its norm by the absolute value of the scalar
  • Triangle inequality: $\|x + y\| \leq \|x\| + \|y\|$, the norm of the sum of two vectors is less than or equal to the sum of their norms

Properties of norm functions

• Positive definiteness:
  • $\|x\| \geq 0$ for all $x \in V$, norms are non-negative
  • $\|x\| = 0$ if and only if $x = 0$, only the zero vector has zero norm
• Homogeneity:
  • $\|\alpha x\| = |\alpha| \|x\|$ for all $x \in V$ and $\alpha \in \mathbb{F}$, scaling a vector by a scalar scales its norm by the absolute value of the scalar
  • Ensures norm is compatible with scalar multiplication in the vector space
• Triangle inequality:
  • $\|x + y\| \leq \|x\| + \|y\|$ for all $x, y \in V$, the norm of the sum of two vectors is less than or equal to the sum of their norms
  • Often proven using properties of the absolute value and the scalar field
  • Geometrically, the shortest path between two points is a straight line

Examples of normed linear spaces

• Euclidean space $\mathbb{R}^n$ with the Euclidean norm:
  • For $x = (x_1, \ldots, x_n) \in \mathbb{R}^n$, $\|x\| = \sqrt{\sum_{i=1}^n |x_i|^2}$
  • Measures the "length" of a vector in $\mathbb{R}^n$
• Space of continuous functions $C[a, b]$ with the supremum norm:
  • For $f \in C[a, b]$, $\|f\| = \sup_{x \in [a, b]} |f(x)|$
  • Measures the maximum absolute value of a function on the interval $[a, b]$
• Space of square-summable sequences $\ell^2$ with the $\ell^2$-norm:
  • For $x = (x_1, x_2, \ldots) \in \ell^2$, $\|x\| = \sqrt{\sum_{i=1}^\infty |x_i|^2}$
  • Measures the "size" of a sequence based on the sum of the squares of its terms

Geometric interpretation of norms

• Norm measures the "length" or "size" of vectors in the space
• Induces a metric $d$ on the space, defined by $d(x, y) = \|x - y\|$ for all $x, y \in V$
  • Measures the "distance" between two vectors in the space
  • Satisfies the following properties for all $x, y, z \in V$:
    1. Non-negativity: $d(x, y) \geq 0$ and $d(x, y) = 0$ if and only if $x = y$
    2. Symmetry: $d(x, y) = d(y, x)$
    3. Triangle inequality: $d(x, z) \leq d(x, y) + d(y, z)$
• Allows for the study of geometric properties of the space
  • Convergence of sequences: a sequence $(x_n)$ in $V$ converges to $x \in V$ if $\lim_{n \to \infty} \|x_n - x\| = 0$
  • Continuity of functions: a function $f: V \to W$ between normed linear spaces is continuous if for every $\varepsilon > 0$, there exists $\delta > 0$ such that $\|f(x) - f(y)\| < \varepsilon$ whenever $\|x - y\| < \delta$