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.
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).