There have been several works characterizing weak topology by nonstandard analysis, which give rise to the following monad ($X$ is a Hilbert space): $$\mu(0) = \{y\in{}^{*}X: \forall x\in X ~~ \langle y|x\rangle\simeq0\}.$$ That is collection of vectors (possibly non standard and not near-standard) that have infinitesimal inner product with all other standard vectors.
However, what can we say about norm of vectors in $\mu(0)$? Is the following statement true? $$\forall y \in \mu(0), \exists r \in \mathbb{R}, |y|<r$$ This means every vector in weak monad has finite norm.
This statement implies uniform boundedness theorem. Uniform boundedness theorem means every set that is weakly bounded is bounded. In nonstandard analysis characterization, that means every standard set that can be absorbed by a weak monad can also be absorbed by ordinary monad. The above statement just says a weak monad is contained in ordinary monad times any infinite number.
If this statement is wrong, can we construct any counter example?
Anderson, Robert M.; Rashid, Salim, A nonstandard characterization of weak convergence, Proc. Am. Math. Soc. 69, 327-332 (1978). ZBL0393.03047.
Henson, C. Ward; Moore, L. C. jun., The nonstandard theory of topological vector spaces, Trans. Am. Math. Soc. 172(1972), 405-435 (1973). ZBL0254.46001.