Paul Garrett [email protected] http://www.math.umn.edu/ g̃arrett/ [This document is http://www.math.umn.edu/ ̃garrett/m/fun/notes 2012-13/05 banach.pdf] 1. Basic definitions 2. Riesz’ Lemma 3. Counter-example: non-existence of norm-minimizing element 4. Normed spaces of continuous linear maps 5. Dual spaces of normed spaces 6. Banach-Steinhaus/uniform-boundedness theorem 7. Open mapping theore...

Lemma 2.19 If C(g) does not hold then for each constant C there exist a database D, , and a fact which can be proved, with the rule R, in the database D, but the proof requires more than C steps. Proof: We proceed in a similar way as in the proof of lemma 2.16, with the following diierences: (i) we no longer assume that the IDB input is empty. Instead, we require that there are the following CO...

in the present paper, we study some properties of fuzzy norm of linear operators. at first the bounded inverse theorem on fuzzy normed linear spaces is investigated. then, we prove hahn banach theorem, uniform boundedness theorem and closed graph theorem on fuzzy normed linear spaces. finally the set of all compact operators on these spaces is studied.

In this paper, we prove that every F ∗ space (i.e., Hausdorff topological vector space satisfying the first countable axiom) can be characterized by means of its “standard generating family of pseudo-norms”. By using the standard generating family of pseudo-norms P, the concepts of P-bounded set and γ-maxpseudo-norm-subadditive operator in F ∗ space are introduced. The uniform boundedness princ...

Fix d ≥ 2 and a field k such that char k d. Assume that k contains the dth roots of 1. Then the irreducible components of the curves over k parameterizing preperiodic points of polynomials of the form z + c are geometrically irreducible and have gonality tending to ∞. This implies the function field analogue of the strong uniform boundedness conjecture for preperiodic points of z + c. It also h...