\( \def\R{\mathbb{R}} \def\N{\mathbb{N}} \def\Z{\mathbb{Z}} \def\Q{\mathbb{Q}} \def\eps{\varepsilon} \def\epsilon{\varepsilon} \renewcommand{\geq}{\geqslant} \renewcommand{\leq}{\leqslant} \)

Chapter 4 - Linear Spaces

4.13 Basic Concepts

Main: Index
Previous: 3.12 Real Functions on Metric and Topological Spaces
Next: 4.14 Convex Sets and Functionals. The Hahn-Banach Theorem


Definition 1

A nonempty set $L$ of elements $x,y,z,\ldots$ is said to be a linear space (or vector space) if it satisfies the following three axioms.

(1) Any two elements $x,y\in L$ uniquely determine a third element $x+y\in L$, called the sum of $x$ and $y$, such that
(a) $x + y = y + x$ (commutativity)
(b) $(x + y) + z = x + (y + z)$ (associativity)
(c) There exists an element $0\in L$, called the zero element with the property that $x + 0 = x$ for every $x\in L$
(d) For every element $x\in L$ there exists an element $-x\in L$, called the negative of $x$ with the property that $x + (-x) = 0$.

(2) Any number $\alpha$ and any element $x\in L$ uniquely determines an element $\alpha x\in L$, called the product of $\alpha$ and $x$, such that
(a) $\alpha(\beta x) = (\alpha\beta)x$
(b) $1\cdot x = x$

(3) The operations of addition and multiplication obey the two distributive laws.
(a) $(\alpha + \beta)x = \alpha x + \beta x$
(b) $\alpha(x + y) = \alpha x + \alpha y$

The points $x$, $y$ are called points or vectors, and $\alpha$, $\beta$ are called scalars.

Definition 2

Two linear spaces $L$ and $L^*$ are said to be isomorphic if there is a one-to-one correspondence $x\leftrightarrow x^*$ between $L$ and $L^*$ which preserves operations, in the sense that $$ x\leftrightarrow x^* ,\quad y\leftrightarrow y^* $$ where $x,y\in L$ and $x^*,y^*\in L^*$, implies $$ x + y\leftrightarrow x^* + y^* $$ and $$ \alpha x \leftrightarrow \alpha x^* $$ for some arbitrary $\alpha$.

Linear Dependence
The elements $x,y,\ldots,w$ of a linear space $L$ are linearly dependent if there exists numbers $\alpha, \beta,\ldots,\lambda$, not all zeros, such that $$ \alpha x + \beta y + \ldots + \lambda w = 0. $$ If no such numbers exists, the points are said to be linearly independent. In an $n$-dimensional linear space, any set of $n$ linearly independent elements if called a basis. Finite $n$-dimensional spaces are covered in linear algebra, while real analysis is more about the study of infinite dimensional spaces.

Given a nonempty subset $L'$ of a linear space $L$, if $L'$ satisfies all the axioms of a linear space, then $L'$ is said to be a subspace of $L$.

Factor spaces
Let $L$ be a linear space and $L'$ a subspace of $L$. The elements $x,y\in L$ belong to the same (residue) class generated by $L'$ if the difference $x-y\in L'$. The set of all such classes is called the factor space or quotient space of $L$ relative to $L'$, denoted by $L|L'$. The dimension of the factor space $L|L'$ is called the codimension of $L'$ in $L$.

Theorem 1

Every factor space $L|L'$ is a linear space.

Theorem 2

Let $L'$ be a subspace of a linear space $L$. Then $L'$ has a finite codimension $n$ if and only if there are linearly independent elements $x_1,\ldots, x_n$ in $L$ such that every element $x\in L$ has a unique representation of the form $$ x = \alpha_1x_1 + \ldots + \alpha_nx_n + y, $$ where $\alpha_1,\ldots,\alpha_n$ are numbers and $y\in L'$.

Linear functionals
A numeric function $f:L\rightarrow\R$ is called a functional on $L$. (A little different from the use in 12.1). A functional $f$ is said to be additive if, $$ f(x + y) = f(x) + f(y) $$ for all $x,y\in L$, and homogeneous if $$ f(\alpha x) = \alpha f(x) $$ for every number $\alpha$. A functional defined on a complex linear space is called conjugate-homogeneous if, $$ f(\alpha x) = \overline{\alpha}f(x) $$ for any $\alpha\in\mathbb{C}$, where $\overline{\alpha}$is the complex conjugate of $\alpha$.

Null Space
Let $f$ be a functional defined on a linear space $L$. Then the set $L_f$ of all elements $x\in L$ such that $$ f(x) = 0 $$ is called the null space of $f$, where $f$ is assumed to be nontrivial, i.e. $f\not= 0$.

Theorem 3

Let $x_0$ be any fixed element of $L - L_f$. Then every element $x\in L$ has a unique representation of the form $$ x = \alpha x_0 + y $$ where $y\in L_f$.

Corollary 1

Two elements $x_1$ and $x_2$ belong to the same class generated by $L_f$ if and only if $f(x_1) = f(x_2)$.

Corollary 2

$L_f$ has codimension 1.

Corollary 3

Two nontrivial linear functionals $f$ and $g$ with the same null space are proportional.

Given a linear space $L$, let $L'\subset L$ be any subspace of codimension 1. Then every class in $L$ generated by $L'$ is called a hyperplane "parallel to $L$". In other words, a hyperplane $M'$ parallel to a subspace $L'$ is the set obtained by subjecting $L'$ to the parallel displacement determined by the vector $x_0\in L$, so that $$ M' = L' + x_0 = \{x\mid x = x_0 + y, y\in L'\}. $$ (Closely related to affine spaces).

Theorem 4

Given a linear space $L$, let $f$ be a nontrivial linear functional on $L$. Then the set $$ M_f = \{x\mid f(x) = 1\} $$ is a hyperplane parallel to the null space $L_f$ of the functional. Conversely, let $M' = L' + x_0$ where $x_0\not\in L'$, be any hyperplane parallel to a subspace $L'\subset L$ of codimension 1 and not passing through the origin. Then there exists a unique linear functional $f$ on $L$ such that $$ M' = \{x\mid f(x) = 1\}. $$

Problem 1



Problem 2



Problem 3



Problem 4



Problem 5



Problem 6