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In mathematics, the closed graph theorem may refer to one of several basic results characterizing continuous functions in terms of their graphs. Each gives conditions when functions with closed graphs are necessarily continuous. A blog post [1] by T. Tao lists several closed graph theorems throughout mathematics.
i=1 → y in Y then y = f(x); [4] set-valued function with a closed graph. If F : X → 2 Y is a set-valued function between topological spaces X and Y then the following are equivalent: F has a closed graph (in X × Y); (definition) the graph of F is a closed subset of X × Y; and if Y is compact and Hausdorff then we may add to this list:
Thus, =; i.e., (,) is in the graph of T. Note, to check the closedness of a graph, it’s not even necessary to use the norm topology: if the graph of T is closed in some topology coarser than the norm topology, then it is closed in the norm topology. [1] In practice, this works like this: T is some operator on some function space.
The following properties are easily checked for a linear operator f : D(f) ⊆ X → Y between Banach spaces: If A is closed then A − λId D(f) is closed where λ is a scalar and Id D(f) is the identity function; If f is closed, then its kernel (or nullspace) is a closed vector subspace of X; If f is closed and injective then its inverse f ...
These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.
In this case, Y is the set of real numbers R with the standard metric d Y (y 1, y 2) = |y 1 − y 2 |, and X is a subset of R. In general, the inequality is (trivially) satisfied if x 1 = x 2. Otherwise, one can equivalently define a function to be Lipschitz continuous if and only if there exists a constant K ≥ 0 such that, for all x 1 ≠ x 2,
Given a function: from a set X (the domain) to a set Y (the codomain), the graph of the function is the set [4] = {(, ()):}, which is a subset of the Cartesian product.In the definition of a function in terms of set theory, it is common to identify a function with its graph, although, formally, a function is formed by the triple consisting of its domain, its codomain and its graph.
Graph of a linear function Graph of a polynomial function, here a quadratic function. Graph of two trigonometric functions: sine and cosine. A real function is a real-valued function of a real variable, that is, a function whose codomain is the field of real numbers and whose domain is a set of real numbers that contains an interval.