What is a Hilbert space?
Q: What is a Hilbert space?
A: A Hilbert space is a mathematical concept that uses the mathematics of two and three dimensions to try and describe what happens in greater than three dimensions. It is a vector space with an inner product structure that allows length and angle to be measured, and it must also be complete for calculus to work.
Q: Who named the concept of Hilbert spaces?
A: The concept of Hilbert spaces was first studied in the early 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. John von Neumann was the one who came up with the name "Hilbert Space".
Q: What are some applications of Hilbert spaces?
A: Hilbert spaces are used in many areas such as mathematics, physics, engineering, functional analysis, partial differential equations, quantum mechanics, Fourier analysis (which includes signal processing and heat transfer), ergodic theory (the mathematical basis of thermodynamics), square-integrable functions, sequences, Sobolev spaces made up of generalized functions, Hardy spaces of holomorphic functions.
Q: Are all normal Euclidean spaces also considered to be Hilbert Spaces?
A: Yes - all normal Euclidean spaces are also considered to be Hilbert Spaces.
Q: How did Hilbert Spaces make a difference to functional analysis?
A: The use of Hilbert Spaces made a big difference to functional analysis by providing new methods for studying problems related to this field.
Q: What type of mathematics does one need knowledge about when working with a Hilbert Space?
A: Vector algebra and calculus are normally used when working with a two-dimensional Euclidean plane or three-dimensional space; however these methods can also be used with any finite or infinite number of dimensions when dealing with a Hilber Space.
