description/proof of that for net with directed index set into finite-dimensional real or complex vectors space with canonical topology, convergence exists if convergences of coefficients w.r.t. constant vectors exist, and then, convergence is expressed with convergences
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of convergence of net with directed index set.
- The reader knows a definition of canonical topology for finite-dimensional real vectors space.
- The reader knows a definition of canonical topology for finite-dimensional complex vectors space.
- The reader knows a definition of Euclidean topological space.
- The reader admits the proposition that for any Hausdorff topological space, any net with directed index set can have only 1 convergence at most.
- The reader admits the proposition that for any finite dimensional real vectors space, the topology defined by the Euclidean topology of the coordinates space based on any basis does not depend on the choice of basis.
- The reader admits the proposition that for any net with any directed index set into any finite-dimensional real or complex vectors space with the canonical topology, the convergence exists if and only if the convergences of the components (with respect to any basis) exist, and then, the convergence is expressed with the convergences.
Target Context
- The reader will have a description and a proof of the proposition that for any net with any directed index set into any finite-dimensional real or complex vectors space with the canonical topology, the convergence exists if the convergences of the coefficients with respect to any constant vectors exist, and then, the convergence is expressed with the convergences.
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
\(d\): \(\in \mathbb{N} \setminus \{0\}\)
\(V\): \(\in \{\text{ the } d \text{ -dimensional } F \text{ vectors spaces }\}\), with the canonical topology
\(J\): \(\in \{\text{ the directed index sets }\}\)
\(n\): \(\in \mathbb{N} \setminus \{0\}\)
\((v_1, ..., v_n)\): \(v_j \in V\), possibly with duplications
\(s\): \(: J \to V, t \mapsto s^j (t) v_j\), \(\in \{\text{ the nets with } J\}\)
//
Statements:
(
\(\forall j \in \{1, ..., n\} (\exists lim s^j)\)
\(\implies\)
\(\exists lim s\)
)
\(\land\)
(
\(\forall j \in \{1, ..., n\} (\exists lim s^j)\)
\(\implies\)
\(lim s = lim s^j v_j\)
)
//
\(s^j: J \to F\) is regarded to be the net with the direct index set with \(F\) regarded as the Euclidean topological space or the complex Euclidean topological space.
2: Note
The point is that \((v_1, ..., v_n)\) is not necessarily any basis for \(V\): especially, \(n\) does not need to equal \(d\).
The reverse does not necessarily hold: for example, when \(J = \mathbb{N} \setminus \{0\}\) and \(s (t) = cos (1 / t) v - cos (1 / t) v\), \(s (t) = 0\), so, \(s\) converges, but \(cos (1 / t)\) does not converge.
Compare with the proposition that for any net with any directed index set into any finite-dimensional real or complex vectors space with the canonical topology, the convergence exists if and only if the convergences of the components (with respect to any basis) exist, and then, the convergence is expressed with the convergences.
3: Proof
Whole Strategy: Step 1: suppose that \(lim s^j\) exists and denote it as \(v^j\); Step 2: see that \(lim s\) exists and equals \(v^j v_j\).
Step 0:
Note that \(V\) and \(F\) are Hausdorff.
When a convergence of \(s\) or \(s^j\) exists, the convergence is unique, by the proposition that for any Hausdorff topological space, any net with directed index set can have only 1 convergence at most.
So, we do not need to worry about the uniqueness of convergences.
Let the topology of \(V\) be defined based on any basis, \(B = \{b_1, .., b_d\}\): in fact, the topology does not depend on the choice of the basis, by the proposition that for any finite dimensional real vectors space, the topology defined by the Euclidean topology of the coordinates space based on any basis does not depend on the choice of basis.
Step 1:
Let us suppose that for each \(j \in \{1, ..., n\}\), \(lim s^j\) exists and denote it as \(v^j\).
Step 2:
Let us see that \(lim s\) exists and equals \(v^j v_j\).
\(v_j = {v_j}^l b_l\).
\(s (t) = s^j (t) v_j = s^j (t) {v_j}^l b_l\).
As \(s^j\) converges, for each \(l \in \{1, ..., d\}\), \(s^j {v_j}^l\) converges to \(v^j {v_j}^l\).
By the proposition that for any net with any directed index set into any finite-dimensional real or complex vectors space with the canonical topology, the convergence exists if and only if the convergences of the components (with respect to any basis) exist, and then, the convergence is expressed with the convergences, \(s\) converges to \(v^j {v_j}^l b_j = v^j v_j\).
So, \(lim s = v^j v_j = lim s^j v_j\).
