Net in $\mathcal{B}^*$ converging to unbounded functional

Question

Let $\varphi$ be an unbounded functional on a Banach space $\mathcal{B}$. Can we always find a net of bounded functionals (i.e. in $\mathcal{B}^*$) converging to $\varphi$ in w*-topology?

Any proof or counterexample?

Answer 1

Let $V$ be the directed set of finite dimensional subspaces of $\mathcal B$. For each $v\in V$ choose a continuous projection onto $v$ and denote it with $P_v$ (this works with Hahn-Banach and uses that $v$ is finite dimensional). Now define a functional $\varphi_v := \varphi\lvert_{v}\circ P_v$. Note that $P_v$ is continuous and $\varphi\lvert_v$ is a linear functional on a finite dimensional space and as such continuous. Then $\varphi_v$ is continuous.

Now consider $x\in \mathcal B$. If $x\in v$ then $\varphi_v(x) = \varphi(P_v(x))=\varphi(x)$. Now for any $v\in V$ there is a $w\in V$ with $w\supseteq v$ and $x\in w$. Further for any $u\supseteq w$ you have $x\in u$ also. This implies then that $$\lim_{v\to\mathcal B}\varphi_v(x) = \varphi(x), $$ and $\varphi_v$ converges pointwise to $\varphi$, which is the weak* convergence.