One Dimensional Normed Linear Space : Completeness and Continuity
Not what you're looking for?
Suppose that E is a one-dimensional normed linear space.
Prove that E is complete and that each linear functional on E is continous.
Purchase this Solution
Solution Summary
Completeness and Continuity are investigated for a One Dimensional Normed Linear Space.
Solution Preview
Solution:
a) E = normed one-dimensional vector space
Let X = vector in E, that means X can be expressed as
X = a*U (1)
where U = one-dimensional basis of E, (a) belongs to the field on which E is defined as vector space (in general, real or complex number)
We can choose the basis so that U = unitary vector, that is ||U|| = 1
In this case, it is easy to see that (a) is just ||X||.
Let's consider now a Cauchy sequence X(n) in E:
||X(m+n) - X(m)|| < eps , ...
Purchase this Solution
Free BrainMass Quizzes
Exponential Expressions
In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.
Graphs and Functions
This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.
Multiplying Complex Numbers
This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.
Probability Quiz
Some questions on probability
Know Your Linear Equations
Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.