Closed linear operator

In functional analysis, a branch of mathematics, a closed linear operator or often a closed operator is a partially defined linear operator whose graph is closed (see closed graph property). It is a basic example of an unbounded operator. The closed graph theorem states that a linear operator f : X \to Y between Banach spaces with full domain X is a closed operator if and only if it is a bounded operator. In practice, many operators are unbounded, but it is still desirable to make them have a closed graph. Hence, they cannot be defined on all of X . To stay useful, they are instead defined on a proper but dense subspace, which still allows approximating any vector and keeps key tools (closures, adjoints, spectral theory) available.

In functional analysis, a branch of mathematics, a closed linear operator or often a closed operator is a partially defined linear operator whose graph is closed (see closed graph property). It is a basic example of an unbounded operator. The closed graph theorem states that a linear operator f : X \to Y between Banach spaces with full domain X is a closed operator if and only if it is a bounded operator. In practice, many operators are unbounded, but it is still desirable to make them have a closed graph. Hence, they cannot be defined on all of X . To stay useful, they are instead defined on a proper but dense subspace, which still allows approximating any vector and keeps key tools (closures, adjoints, spectral theory) available. \operatorname{graph}{\!(f)} = \{(x, f(x)) : x \in \operatorname{dom} f\}. However, one exception to this is the definition of "closed graph". A function f : D \subseteq X \to Y is said to have a closed graph if \operatorname{graph} f is a closed subset of X \times Y in the product topology; importantly, note that the product space is X \times Y and D \times Y = \operatorname{dom} f \times Y as it was defined above for ordinary functions. In contrast, when f : D \to Y is considered as an ordinary function (rather than as the partial function f : D \subseteq X \to Y ), then "having a closed graph" would instead mean that \operatorname{graph} f is a closed subset of D \times Y. If \operatorname{graph} f is a closed subset of X \times Y then it is also a closed subset of \operatorname{dom} (f) \times Y although the converse is not guaranteed in general. Definition: If and are topological vector spaces (TVSs) then we call a linear map a closed linear operator if its graph is closed in . The antonym of "closed" is "unclosed", that is, an unclosed linear operator is a linear operator whose graph is strictly smaller than its closure. Consider the derivative operator f = \frac{d}{d x} where X = Y = C([a, b]) is the Banach space (with supremum norm) of all continuous functions on an interval [a, b]. If one takes its domain D(f) to be C^1([a, b]), then f is a closed operator, which is not bounded. On the other hand, if D(f) is the space C^\infty([a, b]) of smooth scalar valued functions then f will no longer be closed, but it will be closable, with the closure being its extension defined on C^1([a, b]). To show that f is not closed when restricted to C^\infty([a, b]) \to C^\infty([a, b]) , take a function u that is C^1 but not smooth, such as u(x) = x^{3/2} . Then mollify it to a sequence of smooth functions (u_n)_{n \in \N} such that \|u_n - u \|_\infty \to 0 , then \|f(u_n) - u' \|_\infty \to 0 , but (u, u') is not in the graph of f|_{C^\infty([a, b])} . A linear operator f admits a closure if and only if for every x \in X and every pair of sequences x_{\bullet} = (x_i)_{i=1}^{\infty} and y_{\bullet} = (y_i)_{i=1}^{\infty} in \operatorname{D}(f) both converging to x in X , such that both f(x_{\bullet}) = (f(x_i))_{i=1}^{\infty} and f(y_{\bullet}) = (f(y_i))_{i=1}^{\infty} converge in Y , one has \lim_{i \to \infty} f(x_i) = \lim_{i \to \infty} f(y_i) .

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