- #1
bhanukiran
- 5
- 0
hilbert space??
hai,
what is hilbert space ?any important links known to you regarding that?please send some links .
hai,
what is hilbert space ?any important links known to you regarding that?please send some links .
bhanukiran said:hai,
what is hilbert space ?any important links known to you regarding that?please send some links .
A Hilbert space is a mathematical concept that represents an infinite-dimensional vector space, which is complete and equipped with an inner product operation. It is named after the German mathematician David Hilbert and is used in many areas of mathematics and physics, including quantum mechanics and functional analysis.
A Hilbert space is different from other vector spaces because it is infinite-dimensional and complete. This means that it contains an infinite number of elements and all Cauchy sequences within the space converge to a unique limit. Additionally, a Hilbert space has an inner product operation that allows for the calculation of angles and distances between vectors.
Some examples of Hilbert spaces include the space of square-integrable functions, the space of continuous functions on a compact interval, and the space of square-summable sequences. In general, any space that satisfies the properties of an inner product operation and completeness can be considered a Hilbert space.
In quantum mechanics, a Hilbert space is used to describe the state of a quantum system. The vectors in the Hilbert space represent the possible states of the system, and the inner product operation is used to calculate probabilities and expectation values of observables. This allows for the mathematical description of phenomena such as superposition and entanglement.
Hilbert spaces are important in mathematics because they provide a framework for studying infinite-dimensional vector spaces. They have applications in various fields, including functional analysis, partial differential equations, and signal processing. Additionally, the properties of Hilbert spaces make them useful for solving many problems in physics and engineering.