Important Definitions in Functional Analysis

by Gaurav Tiwari

Are you getting started with Functional Analysis? If you are, you must be looking for some of the most important definitions in Functional Analysis. In this article, I have summarized those.

Functional Analysis

Functional analysis is the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense.

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\}$

A 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

  • 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 $1\le p <\infty$

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