Analysis 3: Vector Spaces Part 1

Alright so background information for the Analysis 3 course at Warwick University.

First a good amount of information is needed on Vector Spaces, since this is primarily what this course uses when proving theorems and axioms.

First a Definition.

Definition 1.1: A Vector Space is a set of Vectors, V over a field of Scalars, F with 2 Binary operators that satisfy the following 8 axioms:

1) Associativity of Addition: (u+v)+w = u + (v+w)

2) Commutivity of Addition: u+v=v+u

3) Identity of Addition: There exists an element 0 in V called the zero vector such that v+0=v for all v in V

4) For all v in V there exists an additive inverse w in V such that v+w=0. This additive inverse is denoted -v

5) Distributivity of Scalar Multiplication with respect to vector addition: a(u+v)=au+av for all a in F and u,v in V

6) Distributivity of Scalar Multiplication with respect to scalar multiplication: (a+b)u=au+bu for all a,b in F and u in V

7) Compatibility of Scalar Multiplication with Field Multiplication: a(bv)=(ab)v

8) Identity element of Scalar Multiplication: 1v=v, where 1 denotes the multiplicative identity of the field F.

To analyse Vector spaces in the same way one analyses Number systems one needs a definition of Length, like the absolute value function on |R. So let us define the Norm of a Vector Space.

Definition 1.2: A norm of a Vector Space V over |R is a function ||.|| : V -> |R satisfying the following conditions:

1) Positive Definiteness: ||v|| =0 <=> v=0 belonging in V

2) Non Negativity: for all v in V, ||v||=>0

3) Homogeneity: for all X in |R and v in V || Xv||=|X|.||v||

4) Triangle Inequality: for all v,w in V, ||v+w||<=||v||+||w||

We say (V,||.||) is a normed vector space.

Now, there are various ways we can define a Norm Vector space satisfying these 4 conditions, as long as we are consistent throughout the analysis of a question, 2 such examples are ||.||p and ||.||infinity.

Later Posts will include proper Latex instead of horrid shorthand