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Banach Spaces
Published in J. Tinsley Oden, Leszek F. Demkowicz, Applied Functional Analysis, 2017
J. Tinsley Oden, Leszek F. Demkowicz
Locally Convex Topological Vector Space (Bourbaki). Let V be a vector space and pι $ p_\iota $ , ι∈I $ \iota \in I $ , a family (not necessarily countable) of seminorms satisfying the following axiom of separationA∀u≠0∃κ∈I:pκ(u)>0 $$ A\forall \boldsymbol{u}\ne \mathbf{0}\, \exists \kappa \in I\ :\ p_\kappa (\boldsymbol{u})> 0 $$
Distributions
Published in Fabio Silva Botelho, Functional Analysis, Calculus of Variations and Numerical Methods for Models in Physics and Engineering, 2020
Theorem 10.1.3Concerning the last definition we have the following:σ is a topology in D(Ω).Through σ, D(Ω) is made into a locally convex topological vector space.
Locally Convex Topological Vector Spaces
Published in Kenneth Kuttler, Modern Analysis, 2017
What does this say about Banach spaces? If X is a normed linear space, it is a locally convex topological vector space in which Ψ consists of only one seminorm, the norm of the space. Consider the ball in X′,
Stampacchia variational inequality with weak convex mappings
Published in Optimization, 2018
Let E be a locally convex topological vector space, be its topological dual and the duality pairing, that is the function defined by . Given a convex set X in E and a set-valued mapping , we denote by the graph of T, that is, the set . The Stampacchia variational inequality problem (shortly, (SVIP)) consists in finding a pair such that Of course, problem SVIP can be formulated in a topological vector space E (as in [1]), but in this case, in order that the problem to be non-trivial, we must assume that (according to a theorem of LaSalle [2], this happens if E has a proper open and convex subset).