Preimage linear functional

1. Oct 20, 2013

aaaa202

1. The problem statement, all variables and given/known data
Let f be a linear functional and set A=f-1({0})
Show that A is a closed linear subspace.

2. Relevant 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?

3. The attempt at a solution

2. Oct 20, 2013

Dick

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. Oct 20, 2013

CompuChip

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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