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Preimage linear functional

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  • #1
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Homework Statement


Let f be a linear functional and set A=f-1({0})
Show that A is a closed linear subspace.

Homework Equations


The linearity comes from the fact that if f(a)=0 and f(b)=0 then f(βa+γb)=βf(a)+γf(b)=0
But how do we know it is closed? Do we show every sequence in A is convergent inside A or how do you show closedness for a space like this?

The Attempt at a Solution

 

Answers and Replies

  • #2
Dick
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Homework Statement


Let f be a linear functional and set A=f-1({0})
Show that A is a closed linear subspace.

Homework Equations


The linearity comes from the fact that if f(a)=0 and f(b)=0 then f(βa+γb)=βf(a)+γf(b)=0
But how do we know it is closed? Do we show every sequence in A is convergent inside A or how do you show closedness for a space like this?

The Attempt at a Solution

You can't prove it's closed unless f is continuous. If the vector space is infinite dimensional then you have to assume that, if it's finite dimensional then all linear functionals are continuous.
 
  • #3
CompuChip
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Also, the proof depends on your definition of closed. As you thought, the proof using sequences will work quite well (and you will need the continuity of f because it implies that ##\lim f(a_n) = f(\lim a_n)##).

Informational note (aka side-track): If you know something about topology, there is also a nice characterization of continuous functions which states that the pre-image of a closed set is closed - that would give an even faster proof since {0} is closed.
 

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