**Continuous Dual Space**

When dealing with topological vector spaces, one is typically only interested in the continuous linear functionals from the space into the base field. This gives rise to the notion of the"continuous dual space" which is a linear subspace of the algebraic dual space *V**, denoted *V′*. For any *finite-dimensional* normed vector space or topological vector space, such as Euclidean *n-*space, the continuous dual and the algebraic dual coincide. This is however false for any infinite-dimensional normed space, as shown by the example of discontinuous linear maps.

The continuous dual *V′* of a normed vector space *V* (e.g., a Banach space or a Hilbert space) forms a normed vector space. A norm ||*φ*|| of a continuous linear functional on *V* is defined by

This turns the continuous dual into a normed vector space, indeed into a Banach space so long as the underlying field is complete, which is often included in the definition of the normed vector space. In other words, this dual of a normed space over a complete field is necessarily complete.

The continuous dual can be used to define a new topology on *V*, called the weak topology.

Read more about this topic: Dual Space

### Other articles related to "continuous dual space, dual, continuous, space, spaces":

**Continuous Dual Space**- Double Dual

... In analogy with the case of the algebraic double

**dual**, there is always a naturally defined

**continuous**linear operator Ψ V → V′′ from a normed

**space**V into its

**continuous**double

**dual**V′′, defined ... 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 x ∈ V, however several difficulties arise ...

### Famous quotes containing the words space, continuous and/or dual:

“This moment exhibits infinite *space*, but there is a *space* also wherein all moments are infinitely exhibited, and the everlasting duration of infinite *space* is another region and room of joys.”

—Thomas Traherne (1636–1674)

“There was a *continuous* movement now, from Zone Five to Zone Four. And from Zone Four to Zone Three, and from us, up the pass. There was a lightness, a freshness, and an enquiry and a remaking and an inspiration where there had been only stagnation. And closed frontiers. For this is how we all see it now.”

—Doris Lessing (b. 1919)

“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)