Definitions in Functional Analysis

Gaurav Tiwari July 25, 2014 Math 8,182 views

Linear Space or Vector Space over a field K

Definition: The linear space over a field K is a non-empty set along with a function $+ : X \times X \to X$ called linear/vector addition (or just, ‘addition‘) and another function $ \cdot : K \times X \to X$ called scalar multiplication, such that for all elements $x, y, z, \ldots$ in $X$ and $1, k, l, \ldots$ in $K$ :

$x+y = y+x$
$x+(y+z) = (x+y)+z$
there exists $0 \in X$ such that $x+0=x$
there exists $-x \in X$ such that $x+(-x)=0$
$k \cdot (x+y)=k \cdot x+k \cdot y$
$(k+l) \cdot x = k \cdot x + l \cdot x$
$(kl) \cdot x = k \cdot (l \cdot x)$
$1 \cdot x =x$

Set-Set, Set Element Summation & Products

Let small Roman letters like $x, y, a, b, \ldots$ represent the elements & capital letters like $A, B, X, Y, \ldots$ represent sets, then

$x+F = \{ x+y : y \in F \} $
$ E+F= \{ x+y: x \in E, \, y \in F \}$
$kE= \{ kx : x \in E \}$
$E \times F = \{ (x,y) : x \in E, \, y \in F \}$

Convex Subset of a linear space

A subset E of a linear space X over field K is said to be convex if $rx+(1-r)y \in E$ when $x,y \in E$ and $0<r<1$

Convex Hull of a subset of linear space

For $E \subset X$, the smallest convex subset of linear space X containing $E$ is called the convex hull of $E$, denoted by co(E).

co(E)= $ \{ \displaystyle{\sum_{i=1}^n} r_i x_i : x_i \in E; r_i \ge 0; \displaystyle{\sum_{i=1}^n} r_i=1\}$

Subspace of a linear space

A non-empty subset $Y$ of linear space $X$ over K is said to be a subspace of $X$ over K if $kx+ly \in Y$, whenever $x,y \in Y$ & $k,l \in K$.

Span of a subset of linear space

For a non-empty subset E of linear space X over K, the smallest subspace of X containing E is span(E) defined as

span(E) = $ \{ \displaystyle{\sum_{i=1}^n} k_i x_i : x_i \in E; k_i \in K \}$

This set is called the span of E.

REMARK: When span(E)=X , then we say that E spans X. Also, if span(E)=X and E is a linearly independent set, E is called the Hamel Basis (or basis) of linear space X.

Linear Map

Let X and Y be two linear spaces over K. A linear map from X to Y is a function F : X →Y such that $F(k_1 x_1 + k_2 x_2) = k_1 F(x_1)+ k_2 F(x_2)$ for all $x_1, x_2 \in X, k_1, k_2 \in K$

The subspace $$ R(F):= \{ y \in Y : F(x)=y \, \mathbf{for \, some} \, x \in X \} $$ of Y is called the range space$ of F. While, the subspace $$ Z(F) := \{ x \in X : F(x) = 0 \}$$ of X is called the zero space$ of F.

REMARK:

Whenever Z(F)=X, we write F=0.
dim X= dim R + dim Z

Norm

Let X be a linear space over the field K of real or complex numbers. A norm on X is the function $|| \, || : X \to R$ such that for all $x, y \in X$ and $k \in K$,

$||x|| \ge 0$ with $||x||=0$ if and only if $x=0$
$||x+y|| \le ||x||+||y||$
$||kx|| = |k| ||x||$ where $|k|$ is the modulus of $k$.

A normed space X is a linear space with a norm || || on it.

Examples of Normed Space

A descriptive analysis of following normed spaces will be done in next article:

Spaces $\mathbb{R}^n$ and $\mathbb{C}^n$
Sequence spaces $l^p, l^\infty, c, c_0, c_{00}$ where $1\le p <\infty$
p-integrable function spaces $L^p, L^\infty$ where $1\le p <\infty$