Space Hierarchies

MAT 310 : Linear Algebra

Vector Space

A Vector Space V over a Field F is a set with the following properties:

If X,Y, Z are in V and b,c is in F then

  1. X+(Y+Z) = (X+Y)+Z
  2. X+Y = Y+X
  3. There's O in V such that X+O=X
  4. For All v in V, There's -v such that V+ -v = O
  5. c*(X+Y) = c*X+c*Y
  6. (c+b)*X = c*X+b*X
  7. c*(b*X) = (c*b)*X
  8. 1*X=X

Normed Vector Space

A Normed Vector Space is a Vector Space V with a Norm. A Norm N is a function from V to F with the following properties:

If X,Y are in V and c is in F then

    1. N(X)≥0
    2. N(X)=0 iff X=0
  1. N(c*X) = |c|*N(X)
  2. N(X+Y) ≤ N(X) + N(Y)

Inner Product Space

An Inner Product Space is a Vector Space V with an Inner Product. An Inner Product I is a function from VxV to F with the following properties:

If X,Y, Z are in V and c is in F then

  1. I(X, Y) = I(Y, X)
  2. I(cX, Y) = cI(X, Y)
  3. I(X+Y, Z) = I(X, Z) + I(Y, Z)
    1. I(X, Y)≥0
    2. I(X, X)=0 iff X=0