# Preimage linear functional

## 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

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Dick
Homework Helper

## 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.

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 $\lim f(a_n) = f(\lim a_n)$).