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Q56E

Expert-verifiedFound in: Page 177

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.

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.

$(1,0,0,...),(0,1,00,...),(0,0,1,0,...),...,(0,0,0,...,0,1,0,...,)$.

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.

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