Abstract Outline Preliminaries General setting Applications to operator algebras Applications to LCS’s The triangle of operators, topologies, bornologies ? % Ngai-Ching Wong Department of Applied Mathematics National Sun Yat-sen University Taiwan ¬È . Tàó. http:\\www.math.nsysu.edu.tw\ ∼wong In memory of my teacher Yau-Chuen Wong (1935.10.2–1994.11.7) ??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 1 Grothendieck (via Banach space operators), 2 Randtke (via continuous seminorms) and 3 Hogbe-Nlend (via convex bounded sets) are compared. the topological method, the bornological method. In terms of Pietsch’s operator ideals, we establish the equivalence of the notions of operators, topologies and bornologies. The approaches in the study of locally convex spaces of Abstract Outline Preliminaries General setting Applications to operator algebras Applications to LCS’s Abstract In this talk, we shall discuss two common techniques in functional analysis: ??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 1 Grothendieck (via Banach space operators), 2 Randtke (via continuous seminorms) and 3 Hogbe-Nlend (via convex bounded sets) are compared. the bornological method. In terms of Pietsch’s operator ideals, we establish the equivalence of the notions of operators, topologies and bornologies. The approaches in the study of locally convex spaces of Abstract Outline Preliminaries General setting Applications to operator algebras Applications to LCS’s Abstract In this talk, we shall discuss two common techniques in functional analysis: the topological method, ??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 1 Grothendieck (via Banach space operators), 2 Randtke (via continuous seminorms) and 3 Hogbe-Nlend (via convex bounded sets) are compared. In terms of Pietsch’s operator ideals, we establish the equivalence of the notions of operators, topologies and bornologies. The approaches in the study of locally convex spaces of Abstract Outline Preliminaries General setting Applications to operator algebras Applications to LCS’s Abstract In this talk, we shall discuss two common techniques in functional analysis: the topological method, the bornological method. ??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 1 Grothendieck (via Banach space operators), 2 Randtke (via continuous seminorms) and 3 Hogbe-Nlend (via convex bounded sets) are compared. The approaches in the study of locally convex spaces of Abstract Outline Preliminaries General setting Applications to operator algebras Applications to LCS’s Abstract In this talk, we shall discuss two common techniques in functional analysis: the topological method, the bornological method. In terms of Pietsch’s operator ideals, we establish the equivalence of the notions of operators, topologies and bornologies. ??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 1 Grothendieck (via Banach space operators), 2 Randtke (via continuous seminorms) and 3 Hogbe-Nlend (via convex bounded sets) are compared. Abstract Outline Preliminaries General setting Applications to operator algebras Applications to LCS’s Abstract In this talk, we shall discuss two common techniques in functional analysis: the topological method, the bornological method. In terms of Pietsch’s operator ideals, we establish the equivalence of the notions of operators, topologies and bornologies. The approaches in the study of locally convex spaces of ??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 2 Randtke (via continuous seminorms) and 3 Hogbe-Nlend (via convex bounded sets) are compared. Abstract Outline Preliminaries General setting Applications to operator algebras Applications to LCS’s Abstract In this talk, we shall discuss two common techniques in functional analysis: the topological method, the bornological method. In terms of Pietsch’s operator ideals, we establish the equivalence of the notions of operators, topologies and bornologies. The approaches in the study of locally convex spaces of 1 Grothendieck (via Banach space operators), ??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 3 Hogbe-Nlend (via convex bounded sets) are compared. Abstract Outline Preliminaries General setting Applications to operator algebras Applications to LCS’s Abstract In this talk, we shall discuss two common techniques in functional analysis: the topological method, the bornological method. In terms of Pietsch’s operator ideals, we establish the equivalence of the notions of operators, topologies and bornologies. The approaches in the study of locally convex spaces of 1 Grothendieck (via Banach space operators), 2 Randtke (via continuous seminorms) and ??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Outline Preliminaries General setting Applications to operator algebras Applications to LCS’s Abstract In this talk, we shall discuss two common techniques in functional analysis: the topological method, the bornological method. In terms of Pietsch’s operator ideals, we establish the equivalence of the notions of operators, topologies and bornologies. The approaches in the study of locally convex spaces of 1 Grothendieck (via Banach space operators), 2 Randtke (via continuous seminorms) and 3 Hogbe-Nlend (via convex bounded sets) are compared. ??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Outline Preliminaries General setting Applications to operator algebras Applications to LCS’s 1 Abstract 2 Preliminaries Uniformities Bornologies 3 General setting Mackey-Aren duality Grothendieck’s construction Randtke’s construction Hogbe-Nlend’s construction The diamond The triangle 4 Applications to operator algebras 5 Applications to LCS’s Schwartz spaces and co-Schwartz spaces Nuclear spaces and co-nuclear spaces ??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies But, ... Consider a new metric −1 −1 d1(x, y) = | tan x − tan y| on R. d1 defines the same usual topology on R. Now, the sequence xn = n, n = 1, 2,..., is d1-Cauchy, but not converge. Note that the convergence of a sequence (or net) is a topological property, while it being Cauchy is a metric property. In particular, every Cauchy sequence is convergent. Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s Motivation – completeness Consider R in its usual metric topology. ??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Consider a new metric −1 −1 d1(x, y) = | tan x − tan y| on R. d1 defines the same usual topology on R. Now, the sequence xn = n, n = 1, 2,..., is d1-Cauchy, but not converge. Note that the convergence of a sequence (or net) is a topological property, while it being Cauchy is a metric property. But, ... Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s Motivation – completeness Consider R in its usual metric topology. In particular, every Cauchy sequence is convergent. ??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Note that the convergence of a sequence (or net) is a topological property, while it being Cauchy is a metric property. Consider a new metric −1 −1 d1(x, y) = | tan x − tan y| on R. d1 defines the same usual topology on R. Now, the sequence xn = n, n = 1, 2,..., is d1-Cauchy, but not converge. Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s Motivation – completeness Consider R in its usual metric topology. In particular, every Cauchy sequence is convergent. But, ... ??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Note that the convergence of a sequence (or net) is a topological property, while it being Cauchy is a metric property. d1 defines the same usual topology on R. Now, the sequence xn = n, n = 1, 2,..., is d1-Cauchy, but not converge. Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s Motivation – completeness Consider R in its usual metric topology. In particular, every Cauchy sequence is convergent. But, ... Consider a new metric −1 −1 d1(x, y) = | tan x − tan y| on R. ??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Note that the convergence of a sequence (or net) is a topological property, while it being Cauchy is a metric property. Now, the sequence xn = n, n = 1, 2,..., is d1-Cauchy, but not converge. Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s Motivation – completeness Consider R in its usual metric topology. In particular, every Cauchy sequence is convergent. But, ... Consider a new metric −1 −1 d1(x, y) = | tan x − tan y| on R. d1 defines the same usual topology on R. ??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Note that the convergence of a sequence (or net) is a topological property, while it being Cauchy is a metric property. but not converge. Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s Motivation – completeness Consider R in its usual metric topology. In particular, every Cauchy sequence is convergent. But, ... Consider a new metric −1 −1 d1(x, y) = | tan x − tan y| on R. d1 defines