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Useful definitions for functional analysis

Useful definitions for functional analysis

Normed spaces:  A vector space X is said to be a normed space if to every x∊ X there is associated a nonnegative real number l l x ll, called the norm of x, in such a way that

(a) || x + Y ||  < || x||  + || Y || for all x and y in X,

(b) || αx|| = | α | || x || if x ∊ X and α is a scalar,

(c) || x || > 0 ifx # 0.

Open unit ball and the closed unit ball : In any metric space, the open ball with center at x and radius r is the set,

Br(x) =  { y : d(x, y) < r} .

 

In particular, if X is a normed space, the sets

B1(O) = {x : || x || < 1 } and B1(O) = {x : || x || < 1 }

are the open unit ball and the closed unit ball of X, respectively.

Subspace : A set Y ⊂ X is called a subspace of X if Y is itself a vector space. One checks easily that this happens if and only if 0 ∊ Y and

αY + βY ⊂  Y

for all scalars ,α and β.

Convex: A set C  ⊂   X is said to be convex if

tC + (1 – t)C  ⊂   C (0 < t < 1).

Balanced: A set B ⊂ X is said to be balanced if αB ⊂ B for every  α∊ Ф  with |α| ≤ 1

Topological spaces : A topological space is a set S in which a collection 𝜏 of subsets (called open sets) has been specified, with the following

properties :

(i) S is open, 0 is open, the intersection of any two open sets is open,

and

(ii) The union of every collection of open sets is open. Such a collection 𝜏 is called a topology on S.

Closed set: A set E ⊂ S is closed if and only if its complement is open.

Compact set: A set K⊂ S is compact if every open cover of K has a finite subcover.

Local base:  A collection 𝛾 of neighborhoods of a point p ∊  S is a local base at p if every neighborhood of p contains a member of 𝛾.

Topological vector spaces: Suppose 𝜏 is a topology on a vector space X such that

(a) every point of X is a closed set, and

(b) the vector space operations are continuous with respect to 𝜏

Under these conditions, 𝜏  is said to be a vector topology on X, and X is a topological vector space.

Types of topological vector spaces: In the following definitions, X

always denotes a topological vector space, with topology 𝜏

(a) X is locally convex if there is a local base ℬ whose members are convex.

(b) X is locally bounded if 0 has a bounded neighborhood.

(c) X is locally compact if 0 has a neighborhood whose closure is compact.

(d) X is metrizable if 𝜏  is compatible with some metric d.

(e) X is an F-space if its topology 𝜏 is induced by a complete invariant metric d.

(f) X is a Frechet space if X is a locally convex F-space.

(g) X is normable if a norm exists on X such that the metric induced by the norm is compatible with 𝜏.

(h) X has the Heine-Borel property if every closed and bounded subset of X is compact.

 

 

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