6 Matching Annotations
1. Apr 2023
2. Local file Local file
1. 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. 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.

#### Annotators

3. Mar 2023
4. Local file Local file
1. 2.4-5 Theorem (Equivalent norms).

In a finite dimensional space, every norm is Equivalent

2. 2.4-3 Theorem (Closedness)

Every finite dimensional Banach space is closed.

3. 2.4-2 Theorem (Completeness).

Every finite dimension subspace of the normed space is complete, so are their subspace.

#### Annotators

5. Jul 2022
6. gist.github.com gist.github.com
1. 2.4 Design a plan.

2.4 Design a plan.