By Hahn Banach, there exists a bounded linear functional " ? " such that.
2.
For any bounded linear functional f defined on B, that is, for any f in the dual space B '.
3.
One can deduce this statement about linear functionals from the statement about positive linear functionals by first showing that a bounded linear functional can be written as a finite linear combination of positive ones.
4.
The definition of a random element X with values in a Banach space B is typically understood to utilize the smallest \ sigma-algebra on " B " for which every bounded linear functional is measurable.
5.
Here, a ( \ cdot, \ cdot ) is a bilinear form ( the exact requirements on a ( \ cdot, \ cdot ) will be specified later ) and f is a bounded linear functional on V.
6.
When the sequence in is a weakly Cauchy sequence, the limit above defines a bounded linear functional on the dual, i . e ., an element of the bidual of, and is the limit of in the weak *-topology of the bidual.
7.
In the case of the counting measure on the natural numbers ( producing the sequence space " ! & thinsp; p " } } ), the bounded linear functionals on are exactly those that are bounded on, namely those given by sequences in.
8.
An equivalent definition, in this case, to the above, is that a map X : \ Omega \ rightarrow B, from a probability space, is a random element if f \ circ X is a random variable for every bounded linear functional " f ", or, equivalently, that X is weakly measurable.
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