closed graph theorem


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closed graph theorem

[¦klōzd ¦graf ′thir·əm]
(mathematics)
If T is a linear transformation on Banach space X to Banach space Y whose domain D (T) is closed and whose graph, that is, the set of pairs (x,Tx) for x in D (T), is closed in X × Y, then T is bounded (and hence continuous).
References in periodicals archive ?
In this case, it follows from the closed graph theorem that [M.
The closed graph theorem shows that if B is p-admissible control operator for [(T(t)).
Finally, the closed graph theorem implies that a new necessary condition of p-admissibility of B for [(T(t)).
Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], it is enough, by the closed graph theorem, to show that Range(B) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
In fact, n1 clearly is a projection and n1 : A([0,2[pi]]) [right arrow] A([0,2[pi]]) is continuous by the closed graph theorem and the continuity of n (and analyticity).
The continuity of D and the closed graph theorem for webbed spaces (see [14, 24.