Functional space
I regarded the whole of the function with a certain property as the functional space (かんすうくうかん, function space) as an object of geometric consideration on specific space.
Summary
The functional space contains various properties of the original space in natural form, and "the reconstruction" can do original space from the functional space if it is good space of the identity. The function targeted for the consideration usually shares 終域 like an actual number function and a complex number level function. As 終域 of the function, I will take the algebraic system such as the specific body and ring as needed, but can think that functional space is thereby given a vector space and the structure of the module in the ring beforehand. Even if original space is not an algebraic thing, it is one of the motives to think about functional space that the consideration that used algebraic operation if I move to functional space is enabled. In other words, the property that was not known by then by returning an algebraic property of the functional space to the original space is discovered, and the property of the certain algebraic system knows that it is decided by moving geometric structure of the adversely original space to the functional space, and thinking.
- When think that change functions for a little general representation; "the whole of the representation" from a certain meeting to a meeting with the distinction called a placement meeting or the placement space (it means that functional space is the specific subset of the placement meeting). I cannot expect the algebraic structure in the natural form then because generally operation may not be defined in a range.
In addition, various phases are defined in functional space and form topological space.
- In this case because the which includes the representation that takes a value as topological space and the same space (the domain is topological space again) in the word "function" is convenient, I am often treated like that. Of course whole R of real number and whole C of the complex number are the same topological space with normal phase.
I change by the context of the discussion what kind of phase is treated, but, for example, it is each point convergence phase that considers it to be the prompt shipment topological space of the copy (only for shares of the density of X), and is introduced naturally of additional character and Y to do, or, for example, equality convergence phase can often see distance phase by the L ∞ - norm of the ルベーグ space as an example again and I face each other, and compact open phase in the functional space on the local site compact space makes a function and the variable it again and treats X equally between the placement sky from X to Y (and I think the function to be one variable) and appears as natural phase about a continuity when I move it at the same time.
The concept such as the functional space in the functional space appears in various form, too. For example, the theory of the distribution of the D rack prescribes space to form of the whole distribution as functional space in the functional space, and, for example, the differential calculus form identifies the surface of the manifold with 接空間 which is the upper functional space locally again, and is a function defined on functional space to be called the cotangent space (bud); (the differential calculus form is cutting of the cotangent bunch globally).
Allied item
This article is taken from the Japanese Wikipedia Functional space
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