# Preimage linear functional

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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?

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.