Proof of 'Any Finite Dimensional Unitary Space is Complete'?

In summary, the conversation discusses the example of "any finite dimensional unitary space is complete" in texts on Hilbert spaces. The concept of unitary is defined as a complex (real) vector space with a complex (sesquilinear / real: bilinear) inner product, also known as a pre-Hilbert space or inner product space. If this space is of finite dimension, then the inner product defines an isometric isomorphism on ##\mathbb{C}^n## (resp. ##\mathbb{R}^n\;##) which is complete.
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kent davidge
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In texts treating Hilbert spaces, it's usually given as an example that "any finite dimensional unitary space is complete", but I've found no proof so far and failed prove it myself.
 
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kent davidge said:
In texts treating Hilbert spaces, it's usually given as an example that "any finite dimensional unitary space is complete", but I've found no proof so far and failed prove it myself.
Unitary means we have a complex (real) vector space with a complex (sesquilinear / real: bilinear) inner product, which is also called a pre-Hilbert space or inner product space. If it is of finite dimension, then we the inner product defines an isometric isomorphism on ##\mathbb{C}^n## (resp. ##\mathbb{R}^n\;##) which is complete.
 
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1.

What is the definition of a finite dimensional unitary space?

A finite dimensional unitary space is a mathematical space that has a finite number of dimensions and satisfies the unitary property, which means that the inner product of any two vectors in the space is equal to their complex conjugate. This type of space is commonly used in quantum mechanics and linear algebra.

2.

What does it mean for a finite dimensional unitary space to be complete?

A finite dimensional unitary space is complete if it contains all possible limits of convergent sequences. In other words, any sequence of vectors that approaches a specific vector will also be contained within the space. This is an important property because it ensures that the space is well-defined and can accurately represent the behavior of the system it is modeling.

3.

How is the completeness of a finite dimensional unitary space proven?

The proof of completeness for a finite dimensional unitary space involves showing that every Cauchy sequence (a sequence in which the elements become arbitrarily close together) in the space converges to a point within the space. This can be done using the properties of unitary spaces and basic concepts from real analysis.

4.

What are the practical applications of the completeness of finite dimensional unitary spaces?

The completeness of finite dimensional unitary spaces is crucial in many areas of science and engineering, particularly in quantum mechanics and signal processing. It allows for the accurate representation and manipulation of complex systems, and is often used in the development of new technologies such as quantum computers.

5.

Are there any exceptions or limitations to the completeness of finite dimensional unitary spaces?

In general, finite dimensional unitary spaces are considered to be complete. However, there are some exceptions and limitations, such as when the space is not properly defined or when the vectors are not continuous. Additionally, the completeness of a space may also depend on the specific inner product used. It is important to carefully consider the properties and limitations of a particular space before using it in scientific research or applications.

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