The phrase "dual by" is not a common or correct usage in written English. It is not grammatically correct and may confuse readers. It is possible that the writer intended to use the phrase "dual of" or "divided by" instead. Example sentence with correct usage: The dual of a square is a rectangle with equal length and width. Incorrect usage: The area of a square can be determined by dual by its side length. Correct usage: The area of a square can be determined by multiplying its side length by itself.
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We also denote the weak convergence of a sequence to in the dual by.
Let X be a reflexive Banach space and denote its dual by X'.
In this sense, it is arguably conceptually appropriate to align even Ibn Gabirol's most emphatic sense of the dual "matter-with-form" composition of Intellect (or, relatedly, his emphasis on a grade of matter "prior to" Intellect) with even Plotinus' staunchest insistence on the unity of Intellect (as for Plotinus, the unity of Intellect, as compared with the unity of the One, is dual by nature).
Indeed the role of the user evaluation was dual: by testing a specific prototype it was possible to gain a micro-view and assess the effectiveness of each component of the complex system; by cumulating the result of all the evaluations (in total 43 people were involved) it was possible to build a macro-view of how cross-language retrieval would impact on users and their tasks.
In the same paper [34], they also introduced another kind of dual, namely, the β-dual (see [35] also, where it is called the g-dual by Chillingworth) defined as X β = { a = ( a k ) : ∑ k a k x k converges for all x = ( x k ) ∈ X }.
In the same paper [4], they also introduced another kind of dual, namely, the β-dual (see [5] also, where it is called the g-dual by Chillingworth) defined as X β = { a = ( a k ) ∈ s : ∑ k a k x k converges for all x = ( x k ) ∈ X }.
We denote its norm by and its dual space by.
Moreover, therapeutic hypothermia plays a dual role by promoting infection and by impairing both host defences and usual infectious criteria.
We denote its norm by ∥ · ∥ and its dual space by E*.
Let X be Banach space, we denote its norm by ||·|| and its dual space by X*.
