1. Infinite Dimensional Vector Spaces

Vector spaces which are not finite dimensional are infinite dimensional vector spaces.
That is if is not is not spanned by a finite set.

Example of spaces with dimension

  1. All real polynomial space is a vector space
  2. Real Sequences
  3. Convergent Sequences
  4. Bounded sequences
  5. All sequences with finitely many non - zero terms
Definition (Span of Infinite Set).

Let be an infinite subset of

Note: is finite

Definition (Linear Independence of Infinite Set).

We say is Liner Independent If any finite subset of it is linearly independent.
That is

Note: is finite

Definition (Basis of Infinite Dimension spaces).

An infinite subset of is said to be a basis iff and is linearly independent

Examples

If is the sequence
If is the sequence Infinite
and so on
is the sequence where in place

The infinite set will be the basis of infinite space in 5th example

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