- #1
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I'm trying to understand a paper which uses weak-* topology. (Unfortunately, the paper was given to me confidentially, so I can't provide a link.) My specific question concerns a use of weak-* topology, and interpretation/use of neighborhoods in that topology.
First, I'll summarize the context:
Let ##H## be a complex vector space with Hermitian form (inner product), denoted ##\langle \psi,\phi\rangle = \overline{\langle \phi,\psi\rangle} ~,## (where ##\psi,\phi\in H##). ##H## has a norm induced by the inner product, but is not necessarily complete in norm topology.
Denote by ##H^\times## the vector space of all antilinear functionals on ##H##, i.e., the algebraic antidual of ##H##, and identify ##\psi\in H## with the antilinear functional on ##H## defined by $$\psi(\phi) := \langle \phi,\psi\rangle ~,~~~ (\phi\in H) ~.$$ With this identification, ##H \subseteq H^\times##.
Now give ##H^\times## the structure of a locally convex space with the weak-* topology induced by the family of seminorms ##\|\Psi\|_\phi := |\Psi(\phi)|##, with ##\Psi\in H^\times,\; \phi\in H##.
This much I understand. I also understand (I think) the standard meaning of weak-* topology on ##H^\times##, and the associated pointwise convergence of sequences of elements ##\{\Psi_k\} \to \Psi## in ##H^\times##.
But now the paper says:
I also don't understand why this definition of neighborhoods means that, "as a consequence", ##\phi_\ell \to \phi## iff ##\phi_\ell(\psi) \to \phi(\psi)## for all ##\psi\in H##.
I would have thought that the last bit, i.e., ##\phi_\ell \to \phi## iff ##\phi_\ell(\psi) \to \phi(\psi)## for all ##\psi\in H## is simply a statement of pointwise convergence in ordinary weak-* topology. But I don't get the relevance of the neighborhoods ##U## defined via a restriction to finite numbers of ##\phi_k\in H##.
I sure hope someone can help me out. My knowledge of general topology is proving insufficient for me to figure it out for myself.
First, I'll summarize the context:
Let ##H## be a complex vector space with Hermitian form (inner product), denoted ##\langle \psi,\phi\rangle = \overline{\langle \phi,\psi\rangle} ~,## (where ##\psi,\phi\in H##). ##H## has a norm induced by the inner product, but is not necessarily complete in norm topology.
Denote by ##H^\times## the vector space of all antilinear functionals on ##H##, i.e., the algebraic antidual of ##H##, and identify ##\psi\in H## with the antilinear functional on ##H## defined by $$\psi(\phi) := \langle \phi,\psi\rangle ~,~~~ (\phi\in H) ~.$$ With this identification, ##H \subseteq H^\times##.
Now give ##H^\times## the structure of a locally convex space with the weak-* topology induced by the family of seminorms ##\|\Psi\|_\phi := |\Psi(\phi)|##, with ##\Psi\in H^\times,\; \phi\in H##.
This much I understand. I also understand (I think) the standard meaning of weak-* topology on ##H^\times##, and the associated pointwise convergence of sequences of elements ##\{\Psi_k\} \to \Psi## in ##H^\times##.
But now the paper says:
I understand why this ##U## can be considered to be a neighborhood in the sense of weak-* topology, but the restriction to a "finite" number of ##\phi_k\in H##mystifies me.Thus, ##U\subseteq H^\times## is a neighborhood of ##\Psi\in H^\times## iff there are finitely many ##\phi_k\in H## such that ##U## contains all ##X\in H^\times## with ##|X(\phi_k) - \Psi(\phi_k)| \le 1## for all ##k##. As a consequence, ##\phi_\ell \in H^\times## converges to ##\phi\in H^\times## iff ##\phi_\ell(\psi) \to \phi(\psi)## for all ##\psi\in H##.
I also don't understand why this definition of neighborhoods means that, "as a consequence", ##\phi_\ell \to \phi## iff ##\phi_\ell(\psi) \to \phi(\psi)## for all ##\psi\in H##.
I would have thought that the last bit, i.e., ##\phi_\ell \to \phi## iff ##\phi_\ell(\psi) \to \phi(\psi)## for all ##\psi\in H## is simply a statement of pointwise convergence in ordinary weak-* topology. But I don't get the relevance of the neighborhoods ##U## defined via a restriction to finite numbers of ##\phi_k\in H##.
I sure hope someone can help me out. My knowledge of general topology is proving insufficient for me to figure it out for myself.