I came across the Definition of Bounded linear functionals, that is:
A linear functional $f$ defined on a normed space $X$ that is, $f: X\mapsto\mathbb{C}$ is said to be bounded if $\exists\, M>0$ such that
$\forall x$ $\in X$, $$|f(x)|\le M\|x\|$$
And since, the dual space $X^{*}$ of $X$, is a normed space under the norm, $$\|f\|=\sup_{x\ne 0}\frac{|f(x)|}{\|x\|}$$ Then we have the following inequality $\forall x \in X$: $$|f(x)|\le \|f\|.\|x\|$$
Comparing this with the above definition, can we actually imply that every linear functional is bounded and therefore continuous? This really confuses me because i have seen some unbounded linear functionals defined in Vector spaces.
I would be grateful to anyone who can clear this confusion.
Best Answer
The dual $X^*$ of a normed space is defined to be the space of bounded linear functionals on $X$. Only for these functionals the operator norm is defined, not for all linear functionals in general. (unless the normed space is finite dimensional, in this case indeed every linear functional is bounded)