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Main Question or Discussion Point
Definition of complete space: if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in converges in M. (and from this definition we can define Hilbert Space)
Definition of Hilbert space:
A Hilbert space is a vector space
with an inner product
such that the norm defined by
turns
into a complete metric space
Definition of complete set: If V is inner product space and {e_{k}}_{k∈ℕ} is orthonormal set of vectors then we say that {e_{k}}_{k∈ℕ} is complete if for every u∈V u=∑<e_{k},u>e_{k}
I have several questions:
In infinite dimension inner vector spaces, infinite set of orthonormal vectors is not necessarily complete vector set? (is this true, and if it is, why?)
Are complete sets and complete vector spaces connected in some way? Is every infinite set of orthonormal vectors in complete space (or more precisely in Hilbert space) complete set?
What is the difference between complete set and basis?
Definition of Hilbert space:
A Hilbert space is a vector space
turns
Definition of complete set: If V is inner product space and {e_{k}}_{k∈ℕ} is orthonormal set of vectors then we say that {e_{k}}_{k∈ℕ} is complete if for every u∈V u=∑<e_{k},u>e_{k}
I have several questions:
In infinite dimension inner vector spaces, infinite set of orthonormal vectors is not necessarily complete vector set? (is this true, and if it is, why?)
Are complete sets and complete vector spaces connected in some way? Is every infinite set of orthonormal vectors in complete space (or more precisely in Hilbert space) complete set?
What is the difference between complete set and basis?
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