A counterexample to the closed graph theorem for bilinear maps

A counterexample to the a-‘theorem’

We exhibit a renormalization group flow for a four-dimensional gauge theory along which the conformal central charge a increases. The flow connects the maximally superconformal point of an N = 2 gauge theory with gauge group SU(N+1) and Nf = 2N flavors in the ultraviolet, to a strongly-coupled superconformal point of the SU(N) gauge theory with Nf = 2N massless flavors in the infrared. Our exam...

Some forms of the closed graph theorem

In this paper we shall establish some forms of the closed, graph theorem for locally convex spaces, using the approach of Ptak(l7). Our interest is in classifying pairs of locally convex spaces (E, F) which have the property that every closed graph linear mapping T: E -> F is continuous; if (E,F) has this property then we shall say that (E, F) is in the class *&. \is# is a particular class of l...

A Counterexample to a Theorem of Chase

A Counterexample to an Endpoint Bilinear Strichartz Inequality

The endpoint Strichartz estimate ‖ef‖L2 t L x (R×R) . ‖f‖L2 x (R) is known to be false by the work of Montgomery-Smith [2], despite being only “logarithmically far” from being true in some sense. In this short note we show that (in sharp constrast to the Lpt,x Strichartz estimates) the situation is not improved by passing to a bilinear setting; more precisely, if P, P ′ are non-trivial smooth F...

Banach's Continuous Inverse Theorem and Closed Graph Theorem

The terminology and notation used here have been introduced in the following articles: [3], [4], [2], [15], [11], [14], [1], [5], [13], [12], [19], [20], [16], [7], [17], [8], [18], [9], and [6]. Let X, Y be non empty normed structures, let x be a point of X, and let y be a point of Y . Then 〈x, y〉 is a point of X × Y. Let X, Y be non empty normed structures, let s1 be a sequence of X, and let ...