definition of Schauder basis for normed vectors space
Topics
About: metric space
The table of contents of this article
Starting Context
- The reader knows a definition of normed vectors space.
Target Context
- The reader will have a definition of Schauder basis for normed vectors space.
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
\( V\): \(\in \{\text{ the normed } F \text{ vectors spaces }\}\) with the metric induced by the norm
\(*B\): \(: \mathbb{N} \setminus \{0\} \to V\)
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Conditions:
\(\forall v \in V (! \exists s: \mathbb{N} \setminus \{0\} \to F (v = \sum_{j \in \mathbb{N} \setminus \{0\}} s (j) B (j)))\)
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2: Note
Inevitably, any finite set, \(\{B (j_1), ..., B (j_n)\}\), is linearly independent, because for \(c^1 B (j_1) + ... + c^n B (j_n) = 0\), there is the \(s: j \mapsto 0\) such that \(0 = \sum_{j \in \mathbb{N} \setminus \{0\}} s (j) B (j)))\), but as \(0 \in V\), such an \(s\) is unique, and so, \(c^1 = ... = c^n = 0\), because otherwise, it would constitute another \(s'\) such that \(0 = \sum_{j \in \mathbb{N} \setminus \{0\}} s' (j) B (j)\).
