Dual Space - Continuous Dual Space - Double Dual

Double Dual

In analogy with the case of the algebraic double dual, there is always a naturally defined continuous linear operator Ψ : VV′′ from a normed space V into its continuous double dual V′′, defined by

$Psi(x)(varphi) = varphi(x), quad x in V, varphi in V'. ,$

As a consequence of the Hahn–Banach theorem, this map is in fact an isometry, meaning ||Ψ(x)|| = ||x|| for all x in V. Normed spaces for which the map Ψ is a bijection are called reflexive.

When V is a topological vector space, one can still define Ψ(x) by the same formula, for every xV, however several difficulties arise. First, when V is not locally convex, the continuous dual may be equal to {0} and the map Ψ trivial. However, if V is Hausdorff and locally convex, the map Ψ is injective from V to the algebraic dual V′* of the continuous dual, again as a consequence of the Hahn–Banach theorem.

Second, even in the locally convex setting, several natural vector space topologies can be defined on the continuous dual V′, so that the continuous double dual V′′ is not uniquely defined as a set. Saying that Ψ maps from V to V′′, or in other words, that Ψ(x) is continuous on V′ for every xV, is a reasonable minimal requirement on the topology of V′, namely that the evaluation mappings

$varphi in V' mapsto varphi(x), quad x in V, ,$

be continuous for the chosen topology on V′. Further, there is still a choice of a topology on V′′, and continuity of Ψ depends upon this choice. As a consequence, defining reflexivity in this framework is more involved than in the normed case.

Other articles related to "double dual, dual, double":

USA Swimming - Meets - Dual Meet, Double Dual, Triangular, Quadrangular, Etc.
... A dual meet is a meet where each individual event is scored based on how individual swimmers on a team swim ... In a dual meet, there is almost always a limit to the number of events that a certain person can swim and to the number of swimmers that a certain team can enter ...
Q-Bus - Interfaces
8-line async DMA MUX M3106 DZQ11-M 4-line double-buffered async EIA MUX, with M3106-PA DZQ11-M 4-line double-buffered async MUX with S-box M3107 DHQ11-M 8-line async MUX module, DMA, Q-22 M3108-A ... M68K processor, HD63486 M7676 KXT11-AB Single-board 16-bit dual-height CPU ... GPIB) M7955-AD MSV11-CD 16-Kword 16-bit MOS RAM with on-board refresh M7957 DZV11 4-line double-buffered async EIA MUX M8006-PA 72-bit opto-isolated output module M8007-PA 72-bit opto-isol ...

Famous quotes containing the words dual and/or double:

Thee for my recitative,
Thee in the driving storm even as now, the snow, the winter-day
declining,
Thee in thy panoply, thy measur’d dual throbbing and thy beat
convulsive,
Thy black cylindric body, golden brass and silvery steel,
Walt Whitman (1819–1892)

Today, San Francisco has experienced a double tragedy of incredible proportions. As acting mayor, I order an immediate state of mourning in our city. The city and county of San Francisco must and will pull itself together at this time. We will carry on as best as we possibly can.... I think we all have to share the same sense of shame and the same sense of outrage.
Dianne Feinstein (b. 1933)