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# x** is weak* continuous iff x** Є j(ε).

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Topology

Suppose that Ε is a normed linear space. Let j: Ε→ Ε** be the canonical imbedding and let x** be a linear functional on Ε*.
Then x** is weak* continuous if and only if x** Є j(Ε).

See the attached file.

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Topology

Suppose that Ε is a normed linear space. Let j: Ε→ Ε** be the canonical imbedding and let x** be a linear functional on Ε*.
Then x** is weak* continuous if and only if x** Є j(Ε).

See the attached file for the solution of the problem.

The intersection of any non-empty family of topologies on is a topology on .
And this topology is weaker than all the above family of topologies and stronger than any topology
which is weaker than all these topologies.
This is the greatest lower bound of this family.

Let be a non-empty class of topological spaces and for each let be a mapping of into .
If is given its discrete topology , then all 's are continuous.
We may find other weaker topologies on which also have the property that all 's are continuous.

In this way we can find a unique weakest topology where all 's are continuous.

The weak topology generated by the 's is defined to be the intersection of all topologies on with
respect to each of which all the 's are continuous mappings.

In the same manner , we can construct a unique weakest topology on where all the linear functionals
are continuous.

Let be a linear functional on such that is a weak* continuous.
Hence is one of the linear functionals

But

Since is a normed linear space , the canonical imbedding is isometric
i.e., ,
Hence is reflexive and since any reflexive normed linear space is a Banach space
which implies that is a Banach space.

which implies that

Conversely, suppose that
Since
which implies that is weak* continuous on .

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Note:- 1.Discrete topology:
Let X be any non-empty set, and let the topology be the class of all the subsets of X.
This topology is called the discrete topology on X and the set X with this topology
is called the discrete space.

Thus any topological space whose topology is the discrete topology is called a
discrete space.

2. Reflexive space:
A normed linear space X is said to be reflexive if X may be identified with its second
dual or the bidual (i.e., the strong dual space of the strong dual space )
by the correspondence .
References :-1. G.F.Simmons, Introduction to Topology and Modern Analysis
2.Yosida , Functional analysis
3.S.C.Bose, Functional analysis
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This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

© BrainMass Inc. brainmass.com October 5, 2022, 3:04 pm ad1c9bdddf>
https://brainmass.com/math/geometry-and-topology/canonical-imbedding-normed-linear-space-38049