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Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, inner product, norm, or topology) and the linear functions defined on these spaces and suitably respecting these structures.
In mathematics, particularly in functional analysis, the closed graph theorem is a result connecting the continuity of a linear operator to a topological property of their graph. Precisely, the theorem states that a linear operator between two Banach spaces is continuous if and only if the graph of the operator is closed (such an operator is ...
A function analysis diagram (FAD) is a method used in engineering design to model and visualize the functions and interactions between components of a system or product. It represents the functional relationships through a diagram consisting of blocks, which represent physical components, and labeled relations/arrows between them, which represent useful or harmful functional interactions.
Add the following into the article's bibliography * {{Rudin Walter Functional Analysis|edition=2}} and then add a citation by using the markup
The spectrum of a linear operator that operates on a Banach space is a fundamental concept of functional analysis. The spectrum consists of all scalars λ {\displaystyle \lambda } such that the operator T − λ {\displaystyle T-\lambda } does not have a bounded inverse on X {\displaystyle X} .
In functional analysis, a state of an operator system is a positive linear functional of norm 1. States in functional analysis generalize the notion of density matrices in quantum mechanics, which represent quantum states, both mixed states and pure states. Density matrices in turn generalize state vectors, which only represent pure states.
In functional analysis, a branch of mathematics, the Borel functional calculus is a functional calculus (that is, an assignment of operators from commutative algebras to functions defined on their spectra), which has particularly broad scope. [1] [2] Thus for instance if T is an operator, applying the squaring function s → s 2 to T yields the ...
The Hahn–Banach theorem states: given a linear functional on a subspace of a complex vector space V, if the absolute value of is bounded above by a seminorm on V, then it extends to a linear functional on V still bounded by the seminorm.