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 Apr 2023

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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.
\( \Vert \alpha_1 + \cdots + \alpha_n\Vert \ge c(\alpha_1 + \cdots + \alpha_n) \)

2.44 Definition (Equivalent norms).
Take note that, if we treat the norm as a type of metric, then the conditions for equivalent norm is strictly stronger than the conditions needed for metric, which is stated in convergence of sequences.

 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.

 Jul 2022

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2.4 Design a plan.
2.4 Design a plan.
