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Given an example of a continuous linear transformation $T: (X, ||\cdot||) \to (Y, ||\cdot||)$, where $(X, ||\cdot||)$ and $(Y, ||\cdot||)$ are normed vector spaces with $\sup_{x \in S}||Tx|| \not = \max_{x \in S}||Tx||$ and $S = \{x \in X : ||x|| = 1 \}$

I've thought of some linear transformation, but when I calculus, does not meet requirement.

Some idea?

I appreciate any help.

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I'll try to get you off to a good start.

The first step is to select an appropriate space $X$, in which the unit circle is not compact. This is important because if $S$ is compact in $S$, then the image of $S$ under $T$ is going to be compact since $T$ is continuous. That would be no good for you, since then $||Tx||$ will attain its supremum.

So, think of your favorite space in which the unit circle is not compact and try to start there. e.g. $C([0,1])$, the space of continuous functions on the interval $[0,1]$ with the supremum norm (you can see this is not sequentially compact by considering the sequence of monomials $x^n$, which converges to something discontinuous).

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  • $\begingroup$ Thank for you help, i try to build the example based on your help. $\endgroup$ – jfruizc273 Feb 24 '17 at 6:13

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