Chapter 5: Fundamentals of Normed Linear Spaces

Normed linear spaces are the fundamental objects of study in functional analysis. They possess both the algebraic structure of linear algebra (linear operations) and a topological structure (distance and convergence) established through the introduction of a "norm." This chapter begins with linear spaces, transitions to normed linear spaces, and introduces several classical function spaces.

1. Linear Space

Definition 5.1 (Linear Space)

Let \(E\) be a non-empty set and \(K\) be the field of real numbers \(\mathbb{R}\) or complex numbers \(\mathbb{C}\). \(E\) is called a real (or complex) linear space if it satisfies the following properties:

(i) Additive Group Properties: For any \(x, y, z \in E\), there corresponds a unique sum \(x+y \in E\) such that:

(a) Commutativity: \(x+y = y+x\);
(b) Associativity: \((x+y)+z = x+(y+z)\);
(c) There exists a zero element \(\theta\) such that \(x+\theta = x\) for every \(x \in E\);
(d) For any \(x \in E\), there exists an additive inverse element \(-x\) such that \(x+(-x) = \theta\).

(ii) Scalar Multiplication Properties: For any \(x, y \in E\) and any numbers \(\alpha, \beta \in K\), there corresponds a unique product \(\alpha x \in E\) such that:

(a) \(\alpha(\beta x) = (\alpha\beta)x\);
(b) \(1 \cdot x = x\);
(c) \((\alpha+\beta)x = \alpha x + \beta x\);
(d) \(\alpha(x+y) = \alpha x + \alpha y\).

Elements in a linear space are also called vectors. The addition of elements and the multiplication of numbers with elements are collectively referred to as linear operations. For example, \(C[0,1]\) and \(L^{2}[0,1]\) are linear spaces.

2. Properties of Linear Spaces

The following properties can be derived from the definition of a linear space:

(i) \(0 \cdot x = \theta\).
(ii) \((-1)x = -x\).
(iii) \(\alpha \cdot \theta = \theta\).
(iv) \(\alpha x = \theta\) if and only if \(\alpha = 0\) or \(x = \theta\).

Proof Details

Regarding (i): Since \(2(0 \cdot x) = (2 \cdot 0)x = 0 \cdot x\), subtracting \(0 \cdot x\) from both sides yields:

\[ 0 \cdot x = 2(0 \cdot x) - 0 \cdot x = \theta \]

Regarding (ii): Since \((-1)x + x = (-1 + 1)x = 0 \cdot x = \theta\), by the uniqueness of the inverse element, \((-1)x = -x\).

Regarding (iii):

\[ \alpha \cdot \theta = \alpha(\theta + (-\theta)) = \alpha \theta + \alpha(-\theta) = \alpha \theta - \alpha \theta = \theta \]

Regarding (iv): Sufficiency is obvious. For necessity: if \(\alpha x = \theta\) and \(\alpha \neq 0\), then:

\[ x = 1 \cdot x = \left( \frac{1}{\alpha} \cdot \alpha \right) x = \frac{1}{\alpha} (\alpha x) = \frac{1}{\alpha} \theta = \theta \]

3. Subspace, Span, and Isomorphism

3.1 Subspace

Definition 5.2 (Subspace)

Let \(E_0\) be a subset of a linear space \(E\). \(E_0\) is called a linear subspace of \(E\) if it is closed under the linear operations of \(E\) (i.e., for any \(x, y \in E_0\) and \(\alpha \in K\), then \(x+y \in E_0\) and \(\alpha x \in E_0\)).

A subspace of \(E\) that is not equal to \(E\) itself is called a proper subspace.
Linear Independence: A set \(A \subset E\) is said to be linearly independent if any finite number of elements in \(A\) are linearly independent.

3.2 Subspace Spanned by a Subset (span L)

Definition 5.3 (span L)

Let \(L\) be a non-empty subset of a linear space \(E\). The set consisting of all possible finite linear combinations of elements in \(L\) is called the subspace spanned by \(L\), denoted as \(\text{span } L\):

\[ \text{span } L = \left\{ \sum_{k=1}^{n} c_k x_k \mid c_k \in K, x_k \in L, n \in \mathbb{N} \right\} \]

\(\text{span } L\) is the smallest subspace containing \(L\), and it is the intersection of all subspaces containing \(L\).
For example, in \(L^2[-\pi, \pi]\), the subset \(L = \{ \frac{1}{\sqrt{\pi}} \sin kt, \frac{1}{\sqrt{\pi}} \cos kt \}_{k=0}^{\infty}\) spans the subspace of all trigonometric polynomials.

3.3 Isomorphism of Linear Spaces

Let \(E\) and \(E'\) be linear spaces. \(E\) is said to be isomorphic to \(E'\) if there exists a bijective mapping \(T\) from \(E\) to \(E'\) such that \(T(x+y) = Tx + Ty\) and \(T(ax) = aTx\).

4. Direct Sum

Definition 5.4 (Direct Sum)

Let \(L_1, \dots, L_n\) be subspaces of \(E\). If any element \(x \in E\) can be uniquely expressed as \(x = x_1 + \dots + x_n\) (\(x_k \in L_k\)), then \(E\) is called the direct sum of these subspaces, denoted by \(E = L_1 \oplus \dots \oplus L_n\).

Property: If \(E\) is the direct sum of \(L_1, \dots, L_n\), then any non-zero elements \(x_1, \dots, x_n\) chosen from each respective subspace must be linearly independent.
Construction: Consider the set \(E\) of all ordered tuples \(x = (x_1, \dots, x_n)\) where \(x_k \in L_k\). By defining component-wise addition and scalar multiplication, this set forms a direct sum of the spaces \(L_k\).

5. Normed Linear Space

To introduce topology into a linear space, the structure must be integrated with the linear operations.

Definition 5.5 (Norm)

Let \(E\) be a linear space. If for every element \(x \in E\), there corresponds a real number \(\|x\|\) such that:

(i) Non-negativity: \(\|x\| \ge 0\), and \(\|x\| = 0 \iff x = \theta\);
(ii) Homogeneity: \(\|\alpha x\| = |\alpha| \|x\|\);
(iii) Triangle Inequality: \(\|x+y\| \le \|x\| + \|y\|\).

Then \(E\) is called a normed linear space, and \(\|x\|\) is called the norm of the vector \(x\).

5.1 Distance and Strong Convergence

A normed linear space naturally induces a distance: \(\rho(x, y) = \|x - y\|\).

Convergence in Norm (Strong Convergence): A sequence \(\{x_n\}\) converges to \(x\) if \(\|x_n - x\| \rightarrow 0\).

5.2 Continuity of Linear Operations

(i) The norm \(\|x\|\) is a continuous functional: \(| \|x_n\| - \|x\| | \le \|x_n - x\|\). If \(x_n\) converges in norm, then \(\{x_n\}\) is bounded.
(ii) Continuity of Addition: If \(x_n \to x\) and \(y_n \to y\), then \(x_n + y_n \to x+y\).

\[ \|x_n + y_n - (x+y)\| \le \|x_n - x\| + \|y_n - y\| \rightarrow 0 \]
(iii) Continuity of Scalar Multiplication: If \(\alpha_n \to \alpha\) and \(x_n \to x\), then \(\alpha_n x_n \to \alpha x\).

Proof: Continuity of Scalar Multiplication

\[ \|\alpha_n x_n - \alpha x\| = \|\alpha_n x_n - \alpha_n x + \alpha_n x - \alpha x\| \]

\[ \le |\alpha_n| \|x_n - x\| + |\alpha_n - \alpha| \|x\| \rightarrow 0 \]

6. Classical Examples of Normed Linear Spaces

1. \(\mathbb{R}^n\) and \(\mathbb{C}^n\): For \(x = (\xi_1, \dots, \xi_n)\), the Euclidean norm is defined as:

\[ \|x\| = \sqrt{\sum_{j=1}^n |\xi_j|^2} \]
2. Continuous Function Space \(C[a, b]\): The maximum norm is defined as:

\[ \|x\| = \max_{t \in [a, b]} |x(t)| \]
3. Sequence Space \(l^p\) (\(1 \le p < \infty\)): Consists of sequences satisfying \(\sum_{j=1}^\infty |\xi_j|^p < \infty\). The norm is:

\[ \|x\|_p = \left( \sum_{j=1}^\infty |\xi_j|^p \right)^{\frac{1}{p}} \]

The triangle inequality \(\|x+y\|_p \le \|x\|_p + \|y\|_p\) is precisely the Minkowski inequality.
4. Integrable Function Space \(L^p(F)\) (\(1 \le p < \infty\)): For measurable functions on \(F \subset \mathbb{R}\), the norm is defined as:

\[ \|f\|_p = \left( \int_F |f(t)|^p dt \right)^{\frac{1}{p}} \]
5. Space of Continuously Differentiable Functions \(C^k[a, b]\): Functions possessing continuous derivatives up to order \(k\). The norm is:

\[ \|x\| = \sum_{j=0}^{k} \max_{t \in [a, b]} |x^{(j)}(t)| \]