What is the weak star topology on $L(X,U')$?

Asked Nov 01 '18 at 19:30

Active Nov 01 '18 at 20:16

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The following is taken from Peter D. Lax book Functional Analysis. Exercise 5 page 165.

Let $X,U$ be Banach spaces. I want to define the weak* topology on the set of bounded linear maps from $X$ to the dual of $U$ denoted $\mathcal{L}(X,U')$.

Am i correct in assuming that a sub basis for this topology is given by the sets

$$\{M\in \mathcal{L}(X,U')\mid |(Mx)u-(M_0x)u|<r\}$$

where $x\in X$, $u\in U$, $M_0\in \mathcal{L}(X,U')$ and $r>0$?

EDIT: If my understanding of this is correct, we want the map $T:\mathcal{L}(X,U')\rightarrow \mathbb{R}$ defined by $T(M)=(Mx)u$ to be continuous, for any $x\in X$, $u\in U$. Therefore the inverse image of every open set $(a,b)$ must be open in $\mathcal{L}(X,U')$.

Suppose that $(M_0x)u\in (a,b)$. Choose $r$ such that $B_r[(M_0x)u]\subseteq (a,b)$. This denotes the ball of radius $r$ centred at $(M_0x)u$. Then

$$\{M\in \mathcal{L}(X,U')\mid |(Mx)u-(M_0x)u|<r\}\subseteq T^{-1}\Big((a,b)\Big).$$

On the other hand every set of the above type must be open so this is indeed the smallest topology which makes the above function continuous?

edited Nov 01 '18 at 20:16

asked Nov 01 '18 at 19:30

OgvRubin

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The weak$^*$ topology is defined on dual spaces. Is ${\cal L}(X,U')$ the dual of some topological vector space? – Umberto P. Nov 01 '18 at 19:35
Do you mean there is something wrong with my phrasing? This is from my textbook. – OgvRubin Nov 01 '18 at 19:36
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I'm not sure. Is your topology the weakest one that makes the mapping from ${\cal L}(X,U') \to \mathbb R$ given by $$M \mapsto (Mx)(u)$$ continuous for every $x \in X$ and $u \in U$? – Umberto P. Nov 01 '18 at 20:01
I edited to explain my reasoning – OgvRubin Nov 01 '18 at 20:09
Perhaps I am misremembering but I thought that was called the weak operator topology. – Umberto P. Nov 01 '18 at 20:10
You may be right since the weak* topology is never defined in my book. The exercise in the book is given as: define it. Maybe my approach is wrong then. – OgvRubin Nov 01 '18 at 20:11
Would you mind naming the book and exercise number? – Umberto P. Nov 01 '18 at 20:15
Yes I should have done that from the beginning. I've updated the question. Its Lax book on functional analysis – OgvRubin Nov 01 '18 at 20:16

What is the weak star topology on $L(X,U')$?

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