Closed range theorem

In the mathematical theory of Banach spaces, the closed range theorem gives necessary and sufficient conditions for a closed densely defined operator to have closed range. The theorem was proved by Stefan Banach in his 1932 Théorie des opérations linéaires.

Let X and Y be Banach spaces, T : D(X) → Y a closed linear operator whose domain D(X) is dense in X, and $\scriptstyle{T'}$ its transpose. The theorem asserts that the following conditions are equivalent:

• R(T), the range of T, is closed in Y
• $\scriptstyle{R(T')}$, the range of $\scriptstyle{T'}$, is closed in $\scriptstyle{X'}$, the dual of X
• $R(T) = N(T')^\perp=\{y\in Y | \langle x^*,y\rangle = 0\quad {\text{for all}}\quad x^*\in N(T')\}$
• $R(T') = N(T)^\perp=\{x^*\in X' | \langle x^*,y\rangle = 0\quad {\text{for all}}\quad y\in N(T)\}.$

Several corollaries are immediate from the theorem. For instance, a densely defined closed operator T as above has R(T) = Y if and only if the transpose has a continuous inverse. Similarly, $\scriptstyle{R(T') = X'}$ if and only if T has a continuous inverse.

See also

• Fundamental theorem of linear algebra

References

• Yosida, K. (1980), Functional Analysis, Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 123 (6th ed.), Berlin, New York: Springer-Verlag .

This mathematical analysis–related article is a stub. You can help Wikipedia by expanding it.v · Categories: Functional analysis Theorems in functional analysis Mathematical analysis stubs Wikimedia Foundation. 2010. Игры ⚽ Поможем решить контрольную работу Closed linear span Region (mathematical analysis) Look at other dictionaries: Closed list — Part of the Politics series Electoral methods Single winner …   Wikipedia Newton's theorem of revolving orbits — Figure 1: An attractive force F(r) causes the blue planet to move on the cyan circle. The green planet moves three times faster and thus requires a stronger centripetal force, which is supplied by adding an attractive inverse cube force. The …   Wikipedia Essential range — In mathematics, particularly measure theory, the essential range of a function is intuitively the non negligible range of the function. One way of thinking of the essential range of a function is the set on which the range of the function is most …   Wikipedia Banach–Alaoglu theorem — In functional analysis and related branches of mathematics, the Banach–Alaoglu theorem (also known as Alaoglu s theorem) states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology. [Rudin, section …   Wikipedia Arzelà–Ascoli theorem — In mathematics, the Arzelà–Ascoli theorem of functional analysis gives necessary and sufficient conditions to decide whether every subsequence of a given sequence of real valued continuous functions defined on a closed and bounded interval has a… …   Wikipedia Whitney extension theorem — In mathematics, in particular in mathematical analysis, the Whitney extension theorem is a partial converse to Taylor s theorem. Roughly speaking, the theorem asserts that if A is a closed subset of a Euclidean space, then it is possible to… …   Wikipedia Löwenheim–Skolem theorem — In mathematical logic, the Löwenheim–Skolem theorem, named for Leopold Löwenheim and Thoralf Skolem, states that if a countable first order theory has an infinite model, then for every infinite cardinal number κ it has a model of size κ. The… …   Wikipedia Atkinson's theorem — In operator theory, Atkinson s theorem gives a characterization of Fredholm operators. The theorem Let H be a Hilbert space and L ( H ) the bounded operators on H . The following is the classical definition of a Fredholm operator: a T ∈ L ( H )… …   Wikipedia Hille–Yosida theorem — In functional analysis, the Hille–Yosida theorem characterizes one parameter semigroups of linear operators on Banach spaces satisfying certain continuity restrictions. The theorem is useful for solving certain differential equations such as the… …   Wikipedia Spectrum (functional analysis) — In functional analysis, the concept of the spectrum of a bounded operator is a generalisation of the concept of eigenvalues for matrices. Specifically, a complex number λ is said to be in the spectrum of a bounded linear operator T if… …   Wikipedia 18+ © Academic, 2000-2024 Contact us: Technical Support, Advertising Dictionaries export, created on PHP, Joomla, Drupal, WordPress, MODx. Mark and share Search through all dictionaries Translate… Search Internet Share the article and excerpts Direct link … Do a right-click on the link above and select “Copy Link”