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Let $V$ denote the vector space $C^5[a,b]$ over $R$.

How to show it is infinite dimensional?

I know that we can write:

$C^5[a,b]$ = { $f\in$ $C[a,b]$ : $5$th derivative exists and is continuous}

How to show that there does not exist a linearly independent subset of $V$ which spans V ?

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2 Answers 2

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Hint:

Note that $\mathbb{R}[x] \subset C^5([a,b],\mathbb{R})$ and that $\mathbb{R}[x]$ is not finite dimensional.

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Clearly functions $x^n$, $n=1,2,...$ are in your space. To prove that these functions are linearly independent consider the Wronskian.

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  • $\begingroup$ Yes..I can see they are linearly independent..How does that imply anything? $\endgroup$
    – Gitika
    Jun 18, 2020 at 10:42
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    $\begingroup$ That means that the space is inf. dim. A fin. dim. space cannot contain an infinite l.i. subset. $\endgroup$
    – markvs
    Jun 18, 2020 at 12:27

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