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- Homework Statement
- Prove that the Linear space of all sequences whose limit is zero is infinite dimensional. (And one more).

- Relevant Equations
- Infinite dimension means having an infinite basis.

A space is infinite dimensional when its basis is infinite. But how can I ensure that the basis of the space of all sequences whose limit is zero is infinite?

(After solving that, I would like to have a hint on this very similar problem which says:

(After solving that, I would like to have a hint on this very similar problem which says:

*let V be a Linear space of all continuous functions in the interval ##[-\pi, \pi]##. Let S be the subset of V which contain all those functions which satisfy:*

1. $$ \int_{-\pi}^{\pi} f dt =0$$

2. $$ \int_{-\pi}^{\pi} f \cos t dt =0$$

3. $$ \int_{-\pi}^{\pi} f \sin t dt =0$$

Prove that S is infinite dimensional.)1. $$ \int_{-\pi}^{\pi} f dt =0$$

2. $$ \int_{-\pi}^{\pi} f \cos t dt =0$$

3. $$ \int_{-\pi}^{\pi} f \sin t dt =0$$

Prove that S is infinite dimensional.)