Student Analysis Seminar

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Non-concrete methods are genuinely necessary to build functional-analytic pathologies

Speaker: Robert Trosten, NYU Courant

Location: Warren Weaver Hall 201

Date: Monday, March 9, 2026, 12:30 p.m.

Synopsis:

It's a classical theorem of functional analysis that the Riesz representation theorem for L^p spaces, even \ell^p spaces, fails at the endpoint p = ∞: there are pathological functionals on \ell^∞ which don't arise by testing against an \ell^1 sequence. If you try to build such functionals yourself, you will probably find yourself using some sort of nonconstructive technique, be it the Hahn-Banach theorem or the existence of a nonprincipal ultrafilter on the naturals. What if we don't let ourselves use such non-canonical, non-constructive tools? In this talk I make the case that, without tools such as Hahn-Banach, \ell^1 and \ell^∞ "might as well be reflexive," and that moreover this statement can be turned into an honest theorem.

More precisely, in this talk we discuss the consistency of ZF + DC + "(\ell^∞)* = \ell^1" by way of the Shelah model of ZF + DC + "all sets of reals have the Baire property." Along the way, we will also see that separable Banach spaces are concrete: although we don't have our favorite abstract functional analysis theorems in full generality, many of our favorite results for separable spaces carry over to this more restricted setting. We will also see that by resolving some pathologies of functional analysis in this manner, we are forced to create new ones.

For this talk, we will assume familiarity with the basics of functional analysis, but not mathematical logic.