**Definitions**:Ultimate Space = S

s_{i} = arbitrary partition of S

Δs_{i} = infinitesimal s_{i}, that is limit of s_{i} as it tends to zero.

p_{i} = arbitrary point in S = Δs_{i}

**Problem 1**: s_{i} ⊂ S implies s_{i} is contained in S. True or false?
**Ans: true.**

**Problem 2**: ∑^{∞}s_{i} = S, that is, lim s_{i} = S as *i* tends to infinity. True or False?
**Ans: true. All problem 2 is saying is that the ultimate space S = the sum of its parts. **

**Problem 3**: a continuum formed by the arrangement of p_{i}s in the same direction forms a straight line. Describe the arrangement of p_{i}s that:

(a) forms a horizontal continuum

(b) forms a vertical continuum
**Ans: (i) west-east or east-west arrangement
(ii)north-south or south-north arrangement. **

**Problem 4 **: The continuum of p_{i} is a curve if it is not a straight line. That is, all the points are not arranged in the same direction. The continuum is closed if it ends where it started. suppose a curve C encompasses the space s_{i}:

(i) What is C relative to s_{i}?

(ii) What are the points on C called?

(iii) What are the points in s_{i} called?
**Ans: (i) C is the boundary of s _{i}
(ii) The points on C are the boundary-points of s_{i}
(ii) The points in s_{i} are the interior points of s_{i}. **

**Problem 5 **: what properties must s_{i} have to be a Linear space over a field F (usually the Real number field or the Complex number field)?
**Ans: the space s _{i} must have the following properties to be a Linear space:
(a) The result of the addition of any two elements in s_{i} must be an element in s_{i}
(b) The result of a scalar multiplication of the field F and elements of s_{i} must be an element in s_{i}.
Additional properties for any x, y, z in s_{i} and α, β in F are:
(c) (x + y) + z = x +(y + z)
(d) x + y = y + x
(e) 0 is in s_{i} such that x + 0 =x for all x in s_{i}
(f) For each x in s_{i}, there is an element -x (the additive inverse of x) in s_{i}
such that x + (-x) = 0
(g) α(x + y) = αx + αy
(h) (α + β)x = αx + βy
(i) α(βx) = (αβ)x
(j) 1(x) = x.
The elements of a linear space are called points or vectors and the elements of the field are called scalars. Usually, the field is the real number field or the complex number field.
A linear space is also called a vector space. An example is the vector space V_{n}(R), the n implies that the vectors are n-tuples of real numbers. The scalars are real numbers with addition and scalar multiplication defined as follows:
(α^{1},...,α^{n}) + (β^{1},...,β^{n}) = (α^{1} + β^{1},...,α^{n} + β^{n})
γ(α^{1},...,α^{n}) = (γα^{1},...,γα^{n})
**

**Problem 6**: what do the properties in ans 5(a) thru 5(f) tell us about a linear space?
**Ans: the properties in ans 5(a) thru 5(f) tell us that a linear space is an Abelian group. **

**Problem 7**: the vector space V_{n}(R) is a linear space. What other linear space do you know?
**Ans: the set X of all functions from a non empty set T into a field F such that addition and scalar multiplication is defined as follows:
[f + g](t) = f(t) + g(t) and [αf](t) = αf(t), where, f, g are in X, t in T and α in F.
An m x n matrix is also a linear space. **

**Problem 8**: what is the diferrence between a subspace of a linear space and a linear manifold?
**Ans: no difference. a linear subspace is also called a linear manifold. **

**Problem 9**: what is a quotient space?
**Ans: The linear space X/E is called the quotient space of X modulo E. X is a linear space and E its subspace. **

**Problem 10**: Given that n = 1 in the vector space V_{n}(R), what type of space is V_{1}(R)?
**Ans: The real number line. Consequently, the real number line is a linear space over itself. In essence a 1-dimensional linear space. **

Peter Oye Sagay