Table of Contents
Is every Euclidean space a Hilbert space?
A Euclidean space (or an “inner product space”) is a Hilbert space if it is complete with respect to the norm induced by the inner product. Note. Technically, we need a metric to discuss completeness, not a norm.
What is the difference between a Banach space and a Hilbert space?
Similarly with normed spaces it will be easier to work with spaces where every Cauchy sequence is convergent. Such spaces are called Banach spaces and if the norm comes from an inner product then they are called Hilbert spaces. In other words the metric defined by the norm is complete.
How do you define Hilbert space?
In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. Hilbert spaces serve to clarify and generalize the concept of Fourier expansion and certain linear transformations such as the Fourier transform.
Is Banach space complete?
Definition 4.2 A Banach space is a complete normed linear space.
Is every Banach space a Hilbert space?
An infinite-dimensional space can have many different norms. Hilbert spaces with their norm given by the inner product are examples of Banach spaces. While a Hilbert space is always a Banach space, the converse need not hold. Therefore, it is possible for a Banach space not to have a norm given by an inner product.
What is a non-Euclidean Hilbert space?
A non-euclidean Hilbert space: ℓ 2 ( R), the space of square summable real sequences, with the inner product ( ( x n), ( y n)) = ∑ n = 1 ∞ x n y n Thanks for contributing an answer to Mathematics Stack Exchange!
What is the difference between Hilbert space and Banach space?
Hilbert space: a vector space together with an inner product, which is a Banach space with respect to the norm induced by the inner product A non-euclidean Hilbert space: ℓ 2 ( R), the space of square summable real sequences, with the inner product ( ( x n), ( y n)) = ∑ n = 1 ∞ x n y n
What is the generalization of Hilbert space?
A Hilbert space essentially is also a generalization of Euclidean spaces with infinite dimension. Note: this answer is just to give an intuitive idea of this generalization, and to consider infinite-dimensional spaces with a scalar product that they are complete with respect to metric induced by the norm.
What is the Hilbert space of the dot product?
Every finite-dimensional inner product space is also a Hilbert space. The basic feature of the dot product that connects it with Euclidean geometry is that it is related to both the length (or norm) of a vector, denoted ||x||, and to the angle θ between two vectors x and y by means of the formula