definition of orthonormal subset of vectors space with inner product
Topics
About: vectors space
The table of contents of this article
Starting Context
- The reader knows a definition of inner product on real or complex vectors space.
Target Context
- The reader will have a definition of orthonormal subset of vectors space with inner product.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( F\): \(\in \{\mathbb{R}, \mathbb{C}\}\), with the canonical field structure
\( V\): \(\in \{\text{ the } F \text{ vectors spaces }\}\), with any inner product
\(*S\): \(\subseteq V\)
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Conditions:
\(\forall s_j, s_l \in S ((s_j \neq s_l \implies \langle s_j, s_l \rangle = 0) \land (s_j = s_l \implies \langle s_j, s_l \rangle = 1))\)
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2: Note
\(V\) does not need to be finite-dimensional.
\(S\) does not need to be finite or countable.
