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3.12 Real Functions on Metric and Topological Spaces |

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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

(1) Any two elements $x,y\in L$ uniquely determine a third element $x+y\in L$, called the

(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 $x + (-x) = 0$.

(b) $(x + y) + z = x + (y + z)$ (associativity)

(c) There exists an element $0\in L$, called the

(d) For every element $x\in L$ there exists an element $-x\in L$, called the

(2) Any number $\alpha$ and any element $x\in L$ uniquely determines an element $\alpha x\in L$, called the

(a) $\alpha(\beta x) = (\alpha\beta)x$

(b) $1\cdot x = x$

(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) $\alpha(x + y) = \alpha x + \alpha y$

(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

The elements $x,y,\ldots,w$ of a linear space $L$ are

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

Let $L$ be a linear space and $L'$ a subspace of $L$. The elements $x,y\in L$ belong to the same

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 elementÂ $x\in L$ 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 $y\in L'$.

A numeric function $f:L\rightarrow\R$ is called a

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

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

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 $$ M' = \{x\mid f(x) = 1\}. $$

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