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Linear Functionals - Continuity and Boundedness

  • #1

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!
 

Answers and Replies

  • #2
dextercioby
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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
 

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