ON THE HAHN-BANACH THEOREM

LAWRENCE NARICI
2007 Advanced Courses of Mathematical Analysis II  
I love the Hahn-Banach theorem. I love it the way I love Casablanca and the Fontana di Trevi. It is something not so much to be read as fondled. What is "the Hahn-Banach theorem?" Let f be a continuous linear functional defined on a subspace M of a normed space X. Take as the Hahn-Banach theorem the property that f can be extended to a continuous linear functional on X without changing its norm. Innocent enough, but the ramifications of the theorem pervade functional analysis and other
more » ... es (even thermodynamics!) as well. Where did it come from? Were Hahn and Banach the discoverers? The axiom of choice implies it, but what about the converse? Is Hahn-Banach equivalent to the axiom of choice? (No.) Are Hahn-Banach extensions ever unique? They are in more cases than you might think, when the unit ball of the dual is "round," as for p with 1 < p < ∞, for example, but not for 1 or ∞. Instead of a linear functional, suppose we substitute a normed space Y for the scalar field and consider a continuous linear map A : M → Y . Can A be continuously extended to X with the same norm? Well, sometimes. Unsurprisingly, it depends on Y , more specifically, on the "geometry" of Y : If the unit ball of Y is a "cube," as for Y = (R n , · ∞ ) or Y = real ∞, for example, then for any subspace M of any X, any bounded linear map A : M → Y can be extended to X with the same norm. This is not true if Y = (R n , · p ), n > 1, for 1 < p < ∞, despite the topologies being identical. The cubic nature of the unit ball does not suffice, however-if Y = c 0 , the extendibility dies. This article traces the evolution of the analytic form as well as subsequent developments up to 2004.
doi:10.1142/9789812708441_0006 fatcat:ist2ruscmze4xlj4mii3tofc3e