Theory of the Functional Space

Diaa A Ahmed

e-mail diahmed@yahoo.com

(Washington, DC October 2005)

Functional space is diffirent from Hilbert space; it has global properties that Hilbert space does not have.

In the theory of vector spaces we normally deal with vector product and sum of vectors, and the scalar product of two vectors defines the absolute value of the vectors. There exist a class of more general spaces we are dealing with in functional analysis where two different kind vectors appear in a specific relationship; dual vectors in the sense of Dirac. The scalar product of these dual vectors define an absolute value of a different kind; a functional. In the simple formula of the complete Dirac bracket we have a scalar product of dual ket and; bra vectors,

< A' | A''> = d ( A' , A'')

The complete bracket represents what we call a functional space. Functional transformations connect the dual vectors. The functinal space is the manifold where these dual vectors are connected and being projected as its dual coordiantes. The dual vectors are the manifest aspect of that more general manifold.

Now, we assume that a complete set of ket and a complete set of bra vecors form the two subsets of the fundamental set of the functional space, then we derive the general properties of the set by introducing the inner product within the set.

INNER PRODUCT AND GLOBAL PROPERTIES OF THE FUNDAMENTAL SET

We can still assign to each member of a ket or bra set a unique element of the dual set, we can still assign to the inner product of two dual vectors a unique finite number, and we can still enumerate members of the set and have a compact set. The inner product preserves the properties of Hilbert space even when the kets or the bras are no longer residing in Helbert space.

The Fundamental Set, { < A | , | A > }

Duality y_f = 򠦠(y_f)* y_j d j

Measurability f (y) = ( y_f , y )

Denumerability ( P < x | ) P | y > = < x | y >

Connectedness < p' | X > = h^-? 򦣰60;/FONT> e^-iq'p'/h d q' < q' | X >

Completeness < A' | A''> = ( A' - A'')

Compactness

The characteristic defining the functional space is the existence of these integers. the global properties of the functional space make it radically different from Hilbert space; even when the space of kets or the space of bras is no longer a Hilbert space, the global properties of the functional space preserve.

COMMUTATION RELATIONS AND TOPOLOGICAL STRUCURES

gauge principle and variational principle start from the functional w, S, to derive the local 1-form and 2-form respectively.

Ket and bra sets form the open sets of topology.

Ket and bra sets form an orthomodular lattice.

When there is a group operation, a group structure arise and it encode properties of the logic.

Hilbert space is deterministic, functional space is not.