The completeness of Hilbert Space

In summary, the Riesz-Fischer theorem states that a function space is complete if and only if there exists a square-integrable function in the space.
  • #1
Ed Quanta
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0
Can anyone guide me through or point me to a link of a proof that Hilbert space is complete? I am doing a paper on Hilbert space so I introduced some of its properties and now want to show it is complete.
 
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  • #2
what is your,definition ,of hilbert space? i.e. sometimes one defines "a hilbert space" as a complete inner product space, so it then complete by definition, and then task is to produce some exampels.

the standard oens are all spaces of square integrable functions, so completeness would be found in any real analysis book, or functional analysis book.
 
  • #3
A Hilbert space is defined to be a complete inner product space. You cannot 'prove' that it is complete. Here is a link to a definition:

http://mathworld.wolfram.com/HilbertSpace.html
 
  • #4
A Hilbert space is just a Banach space with an inner product. A Banach space is by definition a complete normed vector space with the metric d(x,y) = ||x-y||

The spaces [tex]L^2[/tex] (the space of square-integrable functions) & [tex]l^2[/tex] (square-summable sequences) are examples of Hilbert spaces, and the [tex]L^p[/tex] spaces are complete by the Riesz-Fischer theorem. Have a look at p.125 of Royden's Real Analysis (what else?!) for the proof.
 
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  • #5
Is Royden's real analysis on the web?
 
  • #6
I doubt it.Does Rudin's book have it...?(The proof that [itex] L^{2}\left(\mathbb{R}\right) [/itex] and its complex counterpart are complete preHilbert spaces).

Daniel.
 
  • #7
I don't know about that but I looked up Riesz-Fischer theorem & found that it was only mentioned twice (& not proved).
 
  • #8
A complete function space is a function set in which no Cauchy sequence of functions in the set converge to limits which are not in the set.

The Riesz-Fischer Theorem identifies the set of "square integrable functions" as a complete function (inner-product) space (a Hilbert Space):

Riesz-Fischer Theorem:

Let the functions [itex]f_1(x),f_2(x),...[/itex] be elements in a function space. If:

[tex]\mathop\lim\limits_{m,n\to\infty}||f_n-f_m||^2\equiv\mathop\lim\limits_{m,n\to\infty}\int_a^b|f_n(x)-f_m(x)|^2dx=0[/tex]

then there exists a "square-integrable function" f(x) to which the sequence [itex]f_n(x) [/itex]converges such that:

[tex]\mathop\lim\limits_{n\to\infty}\int_a^b|f(x)-f_n(x)|^2 dx=0[/tex]

Edit: Thus all Cauchy sequences of square-integrable functions converge to functions which themselves are square-integrable. Makes sense right? If they converged to a function which was not square-integrable, then the set of square integrable functions would not be complete. You know . . . I'm pretty sure that's right. Correct me if it could be said differently.
 
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Related to The completeness of Hilbert Space

1. What is Hilbert Space?

Hilbert Space is a mathematical concept that was developed by David Hilbert in the early 20th century. It is a mathematical structure that is used to study infinite-dimensional vector spaces.

2. How is the completeness of Hilbert Space defined?

The completeness of Hilbert Space refers to the property that all Cauchy sequences in the space converge to a limit within the space. This means that every sequence in the space has a well-defined limit within the space itself.

3. Why is completeness important in Hilbert Space?

Completeness is important in Hilbert Space because it allows for the use of analytical and algebraic techniques to solve problems, rather than relying solely on numerical methods. It also ensures that solutions to problems in Hilbert Space are unique and well-defined.

4. Can Hilbert Space be complete and infinite at the same time?

Yes, Hilbert Space can be both complete and infinite. In fact, most Hilbert Spaces are infinite-dimensional, meaning they have an infinite number of basis vectors. Completeness is a property that is independent of the dimensionality of the space.

5. How does the completeness of Hilbert Space relate to the concept of convergence?

The completeness of Hilbert Space is closely related to the concept of convergence. In a complete space, every Cauchy sequence converges to a limit within the space. This means that the sequence is "converging" to a well-defined point, rather than just getting closer and closer to a limit that may not exist in the space.

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