Linear Functionals - Continuity and Boundedness

In summary, the conversation discusses how to prove that a continuous linear functional is bounded and vice versa. The definition of a linear functional and the condition for a bounded linear functional are mentioned. The individual attempts at a solution involve applying the triangle inequality and using a standard functional analysis approach. It is suggested to refer to Prugovecky's "Quantum Mechanics in Hilbert Space" for a neat proof.
  • #1
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Homework Statement



Prove that a continuous linear functional, [tex] f [/tex] is bounded and vice versa.

Homework Equations



I know that the definition of a linear functional is:
[tex] f( \alpha|x> + \beta|y>) = \alpha f(|x> ) + \beta f( |y> ) [/tex]
and that a bounded linear functional satisfies:
[tex] ||f(|x>)) || \leq \epsilon ||\ |x> || , \ \ \ \epsilon > 0[/tex]

The Attempt at a Solution



I tried the following by letting:

[tex] f(|x>) = \sum a_{i}f( |x_{i}> ) [/tex]
then applying triangle inequality:
[tex] || f(|x>) || \leq \sum||a_{i}|| \ || f( |x_{i}> ) || [/tex]

but now I'm stuck, can someone please help get going in right direction? thanks!
 
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  • #2
This is a standard functional analysis thing which is valid in normed/preBanach spaces and is normally presented in books as a result applicable to functionals when seen as operators from a normed space to C/R seen as normed spaces wrt the modulus.

So the proof for operators goes in 2 steps. First you show that a linear operator continuous at a point is continuous everywhere on the normed space it acts. Then you can show the desired equivalence.

This is neatly proved in Prugovecky's <Quantum Mechanics in Hilbert Space>, Ch.III
 

1. What is a linear functional?

A linear functional is a mathematical mapping from a vector space to its underlying field (usually real or complex numbers) that preserves the properties of linearity, which means it follows the rules of addition and scalar multiplication.

2. How is continuity defined for linear functionals?

A linear functional is said to be continuous if and only if it maps convergent sequences to convergent sequences, meaning that the limit of the functional applied to a sequence is equal to the functional applied to the limit of the sequence.

3. What is the significance of continuity for linear functionals?

Continuity is important for linear functionals because it ensures that small changes in the input result in small changes in the output. This is especially useful in applications where small changes in the input should lead to small changes in the output, such as in optimization problems.

4. How is boundedness defined for linear functionals?

A linear functional is said to be bounded if there exists a positive real number such that the absolute value of the functional applied to any vector is less than or equal to that number multiplied by the norm of the vector.

5. Why is boundedness important for linear functionals?

Boundedness is important for linear functionals because it guarantees that the functional does not grow too quickly, which can lead to undesirable results and make it difficult to analyze or use in applications. A bounded linear functional also ensures that it is well-defined and can be extended to the entire vector space.

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