- #1

- 8

- 0

-Definition of complete space: if every Cauchy sequence of points in

-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

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?

*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

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?