Prove that if $F$ is a linear functional then $F(0) = 0$.

Question

Prove that if $F$ is a linear functional then $F(0) = 0\, $(assume the function u = 0 is in the domain of $F$).

My attempt...

Proof
Let $v,w$ be arbitrary functions in the domain of $F$, then $$F(0) = F(0\cdot v + 0\cdot w) = 0\cdot F(v) + 0\cdot F(w) = 0$$

Is this wrong/too simple?

Answer 1

$\text{ }$ Yes, correct. Also $F(0)=F(0)+F(0) \implies F(0)=0.$ $\text{ }$

Answer 2

It's correct. More simple: let $u$ be in the domain of $F$.

$$F(0) = F(0\cdot u) = 0\cdot F(u) = 0$$

And by the way you can see that $u=0$ must be in the domain of $F$ (i.e. no need to assume this) because it is linear.

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@Umberto P. sorry, saw your comment too late.