 Apr 2023

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is beyond all the dominant terms
index n1 is larger than the indices for all the dominating terms in the sequence.


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Lemma 2.13
The cute formula, generalized version.


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2.41 Lemma (Linear combinations)
Norm of the linear combinations of vectors are, larger than the sum of absolute value of all the scalar weight, by a strictly positive constant. Only for finite dimensional spaces.
\( \Vert \alpha_1 + \cdots + \alpha_n\Vert \ge c(\alpha_1 + \cdots + \alpha_n) \)

11. (Convex set, segment)
Unit Ball in a Normed space is convex

subspaces of a normed spaceX (of any dimension)
I just discovered that the subspaces in vector spaces are very different compare to metric spaces.
 A subspace of a metric space just have to be a metric space.
 A subspace of a vector space will still have to retain the vector space structure. But if it's viewed as a metric space, this doesn't have to be the case.
Also take note that this is talking about any spaces of dimensions.

2.56 Theorem (Continuous mapping)
Continuous mapping preserves compactness in finite dimensional spaces.

2.55 Theorem (Finite dimension)
Compact Closed unit ball in a normed spaces would mean that we have finite dimension.

2.54 F. Riesz's Lemma
Riesz's Lemma, preparing for a theorem about the norm ball in finite dimensional spaces.

2.53 Theorem (Compactness).
compactness is euivalent to closed and boundedness in finite dimensional spaces.

2.52 Lemma (Compactness)

2.51 Definition (Compactness)

 Mar 2023

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2.45 Theorem (Equivalent norms).
In a finite dimensional space, every norm is Equivalent

2.43 Theorem (Closedness)
Every finite dimensional Banach space is closed.

2.42 Theorem (Completeness).
Every finite dimension subspace of the normed space is complete, so are their subspace.

2.32 Theorem (Completion)
Alternative explanations from Walfram Math world: here.
In brief, you can use an isometry map from a banach space to another subspace in a different Banach space such that it's dense.

2.31 Theorem (Subspace of a Banach space).
Similar to 1.47


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Working with Reduoed Costs
A reduced cost is generated from a potential assignment on the vertices of the graph.

 Feb 2023

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Property 2.4.
The \(z(0)\) is the objective function whenever the potential for all the vertices are zero, hence it's the objective of the original problem. In this claim we prove that the difference between the problem with zero porential and the problem with some potential and a "reduced costs" is a constant away, and hence, solving the problem with the reduced costs is the same as solving the original.
Is this some type of primal dual algoithm that we are dong here...

Property 2.5
Properties of Reduced costs network. A reduced cost label is created via a potential label on the vertices of the graph. 1. The reduced costs on a path equals to the costs itself. 2. The sum of reduced costs along any path is the original cost minus the differences between the destination and the source.
It's like a line integer over a conservative field in physics you know, the value we get by subtracting the reduced costs over circle and path gives the differences in the potential. Just like a conservative field in physics.

Working with Residual Networks
Residual Network is a transform on the network given an existing feasible flow.

Property 2.2
Trees of graphs:  A tree on n nodes contains exactly n  1 node.  A tree has at least two leaf nodes  Every two nodes of a tree are connected by a unique path.
