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Every linear functional on a topological vector space (TVS) is a linear operator so all of the properties described above for continuous linear operators apply to them. However, because of their specialized nature, we can say even more about continuous linear functionals than we can about more general continuous linear operators.
Continuous linear operator; Densely defined operator – Function that is defined almost everywhere (mathematics) Hahn–Banach theorem – Theorem on extension of bounded linear functionals; Linear extension (linear algebra) – Mathematical function, in linear algebra
A normal operator on a complex Hilbert space is a continuous linear operator: that commutes with its hermitian adjoint, that is: =. [2] Normal operators are important because the spectral theorem holds for them. Today, the class of normal operators is well understood.
In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem or the Banach theorem [1] (named after Stefan Banach and Juliusz Schauder), is a fundamental result that states that if a bounded or continuous linear operator between Banach spaces is surjective then it is an open map.
The continuous linear functionals on B(H) for the weak, strong, and strong * (operator) topologies are the same, and are the finite linear combinations of the linear functionals (xh 1, h 2) for h 1, h 2 ∈ H.
A linear transformation between topological vector spaces, for example normed spaces, may be continuous. If its domain and codomain are the same, it will then be a continuous linear operator. A linear operator on a normed linear space is continuous if and only if it is bounded, for example, when the domain is finite-dimensional. [18]
A linear operator between normed spaces is continuous if an only if it is bounded. The concept of a bounded linear operator has been extended from normed spaces to all topological vector spaces. Outside of functional analysis, when a function f : X → Y {\displaystyle f:X\to Y} is called " bounded " then this usually means that its image f ( X ...
Intuitively, the continuous operator never increases the length of any vector by more than a factor of . Thus the image of a bounded set under a continuous operator is also bounded. Because of this property, the continuous linear operators are also known as bounded operators.
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