2.4-1 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.
\( \Vert \alpha_1 + \cdots + \alpha_n\Vert \ge c(|\alpha_1| + \cdots + |\alpha_n|) \)
2.4-4 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.
2.4-5 Theorem (Equivalent norms).
In a finite dimensional space, every norm is Equivalent
2.4-3 Theorem (Closedness)
Every finite dimensional Banach space is closed.
2.4-2 Theorem (Completeness).
Every finite dimension subspace of the normed space is complete, so are their subspace.
2.4 Design a plan.
2.4 Design a plan.