All about Mathematical Spaces

Hey readers! In this post we’ll discuss a complicated but mostly used concept in mathematics-‘Spaces’ along with their types such as- inner product space, Vector Space, normed linear space, topological space, Banach Space,etc…

Mathematical Spaces:

Mathematical Spaces are mainly the heart of mathematics., any kind of function or mapping is mentioned & operated in some type of Spaces, such as inner product space, Vector space, Normed linear space, etc… Basically, to understand mathematics, we need to know about one thing that is- “Spaces”. Let us first define Spaces-

Spaces:

Mathematical Spaces are Special type of mathematical structure, which satisfy certain properties, (say operations) based on its constituent elements, that are points or elements.

Most initially, simply space means vector space. let us first know what is vector space, to know this we first see a term linear mapping-

Linear mapping:

A linear operator is a mapping from V to W (T: V→W) which satisfy Additivity And Homogeneity i.e.

  • Additivity : T(v+w) = Tv + Tw; ∀ v∈V & w∈W
  • Homogeneity :T(av) =a(Tv); ∀ v∈V & ‘a’ is a scalar.

Vector Spaces:

Vector Space is a set V Satisfying certain (ten) properties as follows:

  1. Closure with respect to (+): (x+y)∈V; ∀x,y∈V
  2. Associative w.r.to (+) : a(x+y)=ax+ay; ∀x,y∈V & ‘a’ is a scalar.
  3. Commutative w.r.to (+) : x+y= y+x.
  4. Existence of additive identity: ∀ 0∈V such that, x+0= 0+x=x.
  5. Existence of additive inverse: ∀ (-x)∈V such that,                         x+(-x) = (-x)+x = 0.
  6. Closure with respect to (.): (x.y)∈V; ∀x,y∈V
  7. Associative w.r.to (.) : a(x.y)=ax.ay; ∀x,y∈V & ‘a’ is a scalar
  8. Associative w.r.to (.) : x(a.b)=xa.xb; ∀x,∈V & ‘a &b’ are scalar.
  9. Commutative w.r.to (.) : x.y= y.x.
  10. Existence of multiplicative identity: ∀ 1∈V such that, x.1= 1.x=x.

Above is a type of Space, Now we move towards further types of Spaces:

Topological Space:

Topological Space is a type of vector spaces, which we define as;
X be a non-empty set & ℑ be a family of subsets of X satisfying following axioms:

  1. The whole set & null set belongs to family ℑ.(i.e. X & Ø∈ℑ)
  2. The intersection of any two sets in ℑ, is in ℑ. In other words, we can say that, intersection of finite no. of sets in ℑ lies in ℑ. i.e. n{Aa:a=1,2,… }∈ℑ.
  3. Union of finite no. of sets in ℑ lies in ℑ.

If some ℑ satisfies above axioms, then the set (X,ℑ) is the Topological Space.

Normed linear space:

A Linear Space (Vector Space) X together with the norm on X is a normed linear space. i.e, a Vector space satisfying norm axioms which are:

  • ||x||=0 (Positive definite)
  • ||x||=0 ⇒ x=0
  • ||ax||=|a| ||x|| ∀ x∈X & ‘a’ is a scalar  (Homogeneity)
  • ||x+y||=||x||+||y|| ∀ x,y∈X  (Triangle inequality)

Metric Space:

A Linear Space (Vector Space) X operated with the distance over all of its constituent elements, i.e. points, which satisfies metric axioms, is a metric Space.Axioms are as follows:

  • d(x,y)=0 (Positive definite)
  • d(x,y)=0 ⇒ x=0
  • d(x,y)=d(y,x)   (Symmetric property)
  • d(x,y)=d(x,z)+d(z,y)   (Triangle inequality)

The figure below clarify the doubts regarding the spaces–

relations spaces
Spaces relations

 

Banach Space:

There is a simple and easy definition of Banach space is that- Banach Space is a complete Normed linear space.

Completeness means here, all the Cauchy sequences in the normed linear space are convergent in the sequence itself.

Inner product:

To define Inner product space, we have to know about what is inner product.

Inner product on V is a function that maps each pair (u,v) of elements of V into a no. <u,v> in F. It satisfies following properties:

  • Positivity: <v,v>=0, ? v?V.
  • Definiteness:<v,v>=0 if & only if,v=0; ? v?V
  • Homogeneity in first slot: <av,w>=a<v,w>; ? v,w?V &  ? a?F
  • Additivity in first slot: <u+v,w>=<u,w>+<v,w>  ? u,v,w?V
  • Conjugate Symmetry: <v,w>=<w,v>*;  ? v,w?V (* denotes conjugate)

Finally, we define the inner product space as;

Inner Product Space:

Inner product space is a vector space v along with an inner product on v.

The sequence of the mathematical spaces are as likely the figure below :

Space map
Space map

This all about mathematical Spaces- a quick review of definitions, with the properties. Hope this will clarify the ideas of various Spaces. Get more knowledge of Algebra with our posts Basic Topology & AlgebraBasic Topology & Algebra 2 & Basic Abstract Algebra..