Insights Blog
-- Browse All Articles --
Physics Articles
Physics Tutorials
Physics Guides
Physics FAQ
Math Articles
Math Tutorials
Math Guides
Math FAQ
Education Articles
Education Guides
Bio/Chem Articles
Technology Guides
Computer Science Tutorials
Forums
General Math
Calculus
Differential Equations
Topology and Analysis
Linear and Abstract Algebra
Differential Geometry
Set Theory, Logic, Probability, Statistics
MATLAB, Maple, Mathematica, LaTeX
Trending
Featured Threads
Log in
Register
What's new
Search
Search
Search titles only
By:
General Math
Calculus
Differential Equations
Topology and Analysis
Linear and Abstract Algebra
Differential Geometry
Set Theory, Logic, Probability, Statistics
MATLAB, Maple, Mathematica, LaTeX
Menu
Log in
Register
Navigation
More options
Contact us
Close Menu
JavaScript is disabled. For a better experience, please enable JavaScript in your browser before proceeding.
You are using an out of date browser. It may not display this or other websites correctly.
You should upgrade or use an
alternative browser
.
Forums
Mathematics
Linear and Abstract Algebra
Complete sets and complete spaces
Reply to thread
Message
[QUOTE="andrewkirk, post: 6020321, member: 265790"] This contains a critical ambiguity - that you have not specified whether the sum is finite or infinite. An orthonormal basis for a Hilbert space H is an orthonormal set of vectors whose span is [I]dense in[/I] H. An orthonormal [I]Hamel[/I] basis for a Hilbert space H is an orthonormal set of vectors whose span is [I]equal to[/I] H. Say ##\{e_k\}_{k\in\mathbb N}## is a basis. Then the infinite, orthonormal subset ##\{e_{2k}\}_{k\in\mathbb N}## of even-numbered elements of the set, is not complete (because if it were the original set would not be linearly independent), and hence not a basis. As I learned it, completeness of a set of vectors does not entail linear independence. If a complete set is not linearly independent, it is not a basis. But your definition above includes orthonormality, which entails linear independence, so your definition looks the same as that of either an orthonormal Hamel basis (if the sums must be finite) or an orthonormal basis (if they can be infinite). [/QUOTE]
Insert quotes…
Post reply
Forums
Mathematics
Linear and Abstract Algebra
Complete sets and complete spaces
Back
Top