Vector Spaces Part 2

Like we left off there are 2 simple types of norms we will use when analysing vector spaces

Proposition 1.3:

1.

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

Mathematics Revision Blog

This blog is going to cover my endeavours to revising for the 4 exams I am taking this summer on Analysis, Vector Analysis, Differentiation and Algebra. These are going to be structured for my benefit only, maybe I'll put some tags and whatever on the blog later idk, might help some other people but this is probably the best way for me to actually learn the subjects and revise succinctly in the time allotted.

Mostly going to concentrate on analysis and vector analysis initially as these subject exams are earlier. Blah blah :)))