smati said:Homework Statement
T1, T2 and T3 are unbounded operators.
What is this condition?
http://T[SUB]1[/SUB]
3. The Attempt at a Solution [/B]
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An unbounded operator is a mathematical object that maps one mathematical space to another, where the two spaces are not necessarily finite-dimensional. In other words, it is a function that takes in a mathematical object and produces another mathematical object as its output.
A bounded operator is one that maps a bounded set to another bounded set, while an unbounded operator is one that can map a bounded set to an unbounded set. Bounded operators are easier to work with mathematically and can often be studied using techniques from linear algebra, while unbounded operators require more advanced techniques from functional analysis.
One common example of an unbounded operator is the derivative operator in calculus. It maps a function to its derivative, and can take in a bounded function but produce an unbounded function as its output. Another example is the Laplace operator in partial differential equations, which maps a function to its second derivative with respect to its independent variables.
The condition in unbounded operators refers to the properties that the operator must satisfy in order to be well-defined and useful in mathematical analysis. This condition is important because it ensures that the operator is not too "wild" or unpredictable, and allows us to make meaningful statements and calculations about the operator and its associated spaces.
Unbounded operators are used in a variety of fields in scientific research, including physics, engineering, and mathematics. They are particularly useful for describing and analyzing systems that involve infinite-dimensional spaces, such as quantum mechanics, fluid dynamics, and signal processing. Unbounded operators also play a key role in the study of differential equations and in functional analysis, which has many applications in modern science and technology.