## Topological vector spaces that are isomorphic to their duals

### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

After reviewing the (locally convex) topological vector spaces that I know, the only examples I could find where there is an isomorphism from the space to its (anti)dual, are Hilbert spaces. So my question is :

Are there topological vector spaces $V$ such that the topology does not come from a Hilbert structure, and such that there exists an isomorphism $\chi : V \to V'$, where $V'$ denotes the antidual of $V$ (continuous antilinear forms on $V$) ?

-

## 3 Answers

An interesting family of examples comes from number theory (or algebraic geometry, depending on who you ask): If you have a field $k$, the Laurent power series field $k((t))$ has an ultrametric topology where `$\{ t^n k[[t]] \}_{n \in \mathbb{Z}}$` form a neighborhood basis of zero. This space is isomorphic to its topological dual: by adjoining $(dt)^{1/2}$, one obtains the perfect residue pairing $$\langle f(t) (dt)^{1/2}, g(t) (dt)^{1/2} \rangle = \operatorname{Res} f(t)g(t) dt.$$ You can do a similar trick with finite dimensional vector spaces over $k((t))$.

If you want nontrivial antilinearity, you may choose $k$ to be a separable quadratic extension of some underfield $F$, and change the residue pairing to be sesquilinear: $$\langle f(t) (dt)^{1/2}, g(t) (dt)^{1/2} \rangle = \operatorname{Res} f(t) \bar{g}(t) dt.$$

-

### You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.

`If $X$ is any reflexive space, then`$X \oplus X^{*}$```is isomorphic to its dual
$X^* \oplus X^{**}$```.

-

Take a reflexive TVS $V$, and consider $V \times V^\ast$.

-

2
Well, technically the question wanted antidual so you need, I guess, an involution on $V$ (an antilinear continuous linear bijection from $V$ to itself). But, yeah, examples abound... – Matthew Daws Feb 5 2012 at 19:50
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 16, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9139239192008972, "perplexity_flag": "head"}