consider the normed linear spaces $(\mathcal C[0,1], ||.|| _i)$.what can you conclude about the correspoding open unit balls?

Asked Dec 22 '14 at 16:55

Active Dec 22 '14 at 17:01

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consider the normed linear spaces $$(\mathcal C[0,1], ||.|| _1), \;(\mathcal C[0,1], ||.|| _2),\;(\mathcal C[0,1], ||.|| _3)\ldots, (\mathcal C[0,1], ||.|| _p)$$ and $(\mathcal C[0,1], ||.|| _\infty)$. Then what can you conclude about the correspoding open unit balls?

My work

$||f||_1=\int_0^1 |f| d\mu\leq1$ (because it is in unit ball)

And so, I can give large height triangle whose area is simply $<1$. So, the unit ball with respect to $||.||_1$ need not be the subset of the unit ball w.r.to $||.||_\infty$ What happens for the remaining cases? If I use $U_i$ for the unit ball w.r.to $||.||_i$, can I conclude that

$$U_1\supset U_2\supset\ldots \supset U_\infty$$

edited Dec 22 '14 at 17:01

asked Dec 22 '14 at 16:55

David

4,201

No $|f|_1 = \int_0^1 |f|d\mu$. You can't forget the absolute value inside the integral, or else $|f|_1=0$ is possible for $f(x)=x-\frac{1}{2}$. – Thomas Andrews Dec 22 '14 at 17:00
yes.. you are right.. thanks for the note. I have corrected.@ThomasAndrews – David Dec 22 '14 at 17:02
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Yes, you can conclude that. Do you know Hölders inequality? – PhoemueX Dec 22 '14 at 23:05
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http://math.stackexchange.com/questions/66029/lp-and-lq-space-inclusion – Rudy the Reindeer Dec 22 '14 at 23:11

consider the normed linear spaces $(\mathcal C[0,1], ||.|| _i)$.what can you conclude about the correspoding open unit balls?

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