Show that an orthonormal(ON) sequence is also a ON-basis in a Hilbert Space

In summary, the problem is to show that a given ON-sequence (f_n) in a Hilbert Space H, with ||e_n - f_n|| < 1, is actually a complete ON-sequence and therefore an ON-basis for H. To solve this, we must consider the subspace generated by (f_n) and how vectors in the orthogonal complement of this subspace expand in terms of the original ON-basis (e_n). By understanding this relationship, we can show that (f_n) is indeed a complete ON-sequence and thus an ON-basis for H.
  • #1
gothlev
3
0
1. Problem description
Let [itex](e_n)_{n=1}^{\infty} [/itex] be an orthonormal(ON) basis for H (Hilbert Space). Assume that [itex] (f_n)_{n=1}^{\infty} [/itex] is an ON-sequence in H that satisfies [itex] \sum_{n=1}^{\infty} ||e_n-f_n|| < 1 [/itex]. Show that [itex](f_n)_{n=1}^{\infty}[/itex] is an ON-basis for H.

Homework Equations



The Attempt at a Solution


Somehow if it can be shown that [tex](f_n)_{n=1}^\infty [/tex] is an complete ON-sequence it can be concluded that [tex](f_n)_{n=1}^\infty [/tex] is a ON-basis for H. I tried to make use of Parseval's formula and also expanding the sum [tex] \sum_{n=1}^\infty ||e_n-f_n|| < 1 [/tex] with the rules for inner products, but it did not really get me anywhere. Since I can not really think of anything else I would need someone to point me in the right direction. I might be missing something really obvious, but can not really see it.
 
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  • #2
You know that [tex](f_n)[/tex] is an orthonormal sequence, so the only way it can fail to be an orthonormal basis is if the closed subspace [tex]V[/tex] generated by [tex](f_n)[/tex] is not the entire space [tex]H[/tex]. Think about how vectors in the orthogonal complement [tex]V^\perp = H \ominus V[/tex] expand in terms of [tex](e_n)[/tex] and how that relates to the [tex]f_n[/tex].
 

Related to Show that an orthonormal(ON) sequence is also a ON-basis in a Hilbert Space

What is an orthonormal sequence?

An orthonormal sequence is a set of vectors in a Hilbert space that are all mutually orthogonal (perpendicular) to each other and have a unit length of 1.

What is an orthonormal basis in a Hilbert space?

An orthonormal basis is a set of vectors in a Hilbert space that form a complete and orthonormal system, meaning that any vector in the space can be written as a unique linear combination of the basis vectors.

How do you show that an orthonormal sequence is also an orthonormal basis in a Hilbert space?

To show that an orthonormal sequence is also an orthonormal basis in a Hilbert space, we must prove two things: (1) the sequence is indeed an orthonormal set and (2) the sequence spans the entire space. This can be done by using the Gram-Schmidt process to show that the sequence is orthogonal and then using the completeness property of Hilbert spaces to show that the sequence spans the space.

What is the Gram-Schmidt process?

The Gram-Schmidt process is a mathematical procedure used to orthogonalize a set of vectors in a vector space. It involves taking a set of linearly independent vectors and creating an orthogonal set of vectors with the same span.

Why is an orthonormal basis important in a Hilbert space?

An orthonormal basis is important in a Hilbert space because it provides a convenient way to represent any vector in the space as a linear combination of the basis vectors. This makes it easier to perform calculations and solve problems in the space. Additionally, an orthonormal basis allows us to define an inner product, which is essential for many applications in mathematics and physics.

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