# Topological Vector Space : Closed Kernel

Not what you're looking for?

Suppose that T is a topologocal vector space. Prove that a linear functional f on T is continuous if and only if ker(f) is closed.

##### Purchase this Solution

##### Solution Summary

It is proven that a linear functional f on a toplogical vector space is continuous if and only if ker(f) is closed.

##### Solution Preview

let f: X ---> R be a non-zero functional from X, a normed

space (any topological vector space will do).

The following are equivalent:

1) f is continuous.

2) the null space (or kernel) N(f) is closed.

3) N(f) is not dense in X.

4) f is bounded on some neighbourhood of 0.

1) --> 2): this is easy, as N(f) = f^-1[{0}] and {0}

is a closed set in ...

##### Purchase this Solution

##### Free BrainMass Quizzes

##### 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.

##### Geometry - Real Life Application Problems

Understanding of how geometry applies to in real-world contexts

##### Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.

##### Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.