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Found in: Page 177

### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

# Show that the space of infinite sequence of real numbers is infinite dimensional.

The solution is the space V of infinite sequence is infinite dimensional.

See the step by step solution

## Step1: Explanation of the solution

Assume that the opposite of that the space V of infinite sequences is finite dimensional with as follows.

dim(V)=n.

Consider an example of n+1 linearly independent infinite sequence as follows.

$\left(1,0,0,...\right),\left(0,1,00,...\right),\left(0,0,1,0,...\right),...,\left(0,0,0,...,0,1,0,...,\right)$.

## Step 2: Draw the conclusion

This contradicts the fact that n is the largest possible dimension for an n-dimensional space that m=n.

Therefore, the assumption is wrong.

Thus, the space V of infinite sequence is infinite dimensional.