Functionals in Reflexive Banach space

Question

I came across a Proposition which stated,

Let $X$ be a Reflexive Banach space, then Every functional $f$ attains it's Norm on X.

We know since Every linear functional defined on a Normed space need not be bounded, but every functional in $X^{*}$ (The Dual space of $X$) is bounded. So if the given Normed space $X$ is reflexive, then by the above Proposition, Every functional attains it's norm. Thus can we say that every linear functional attains it's norm? (Since $|f(x)|\le||f||.||x|| $ $\forall x \in X$) in Other words, can we imply that $X^{*}$ contains All the linear functional defined on X?

Answer 1

You can't really say an unbounded linear functional can "attain its norm".
I guess you could argue that the norm of such a functional would be $+\infty$, in a way, but then you do not have any vector of norm $+\infty$?

Moreover, there always exists an unbounded linear functional in any infinite-dimensional normed space $X$, by taking any infinite family $(x_n)_n$ with linearly independent elements of norm $1$ and assigning $n$ to $x_n$, and assigning $0$ to all vectors of an algebraic complement of $\operatorname{Span}(\{x_n, n \in \mathbb{N}\})$ in $X$. As such, even when $X$ is reflexive, $X^*$ does not contain all the linear functionals of $X$ if $X$ is infinite-dimensional.

The proposition you cited is true, but only makes sense if "every functional" is replaced by "every continuous linear functional" (see here for example: continuous linear functional on a reflexive Banach space attains its norm).