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Consider $C_{\mathbb{F}}[a,b]$ and $C_{\mathbb{F}}[c,d]$ (the space of continuous functions from an interval to the field) for two intervals $[a,b]$ and $[c,d]$, where $-\infty<a<b<\infty$ and $-\infty<c<d<\infty$. Both are supplied with the supremum norm $||\cdot||_\infty$. I have to show that these two Banach spaces are isometrically isomorphic.

I think I have to give an isometry $T: C_{\mathbb{F}}[a,b]\to C_{\mathbb{F}}[c,d]$? I am a bit new to this concept so a hint would be appreciated.

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Given a function $f\in C_\mathbb{F}[a,b]$, we need to specify a function $Tf\in C_\mathbb{F}[c,d]$. To see how we might try this, let's study how we can identify $[c,d]$ with $[a,b]$.

We wish to define a function $G:[c,d]\to[a,b]$ whose graph is a line segment containing the points $(c,d)$ and $(a,b)$. This gives $$ G(x) = a+\frac{b-a}{d-c}(x-c). $$

Using this $G$, let's define $Tf:[c,d]\to\mathbb{F}$ by $(Tf)(t)=f(G(t))$. Since $G$ is a continuous surjection, we have that $Tf=f\circ G$ is continuous and $$ \|Tf\|_\infty=\sup_{t\in[c,d]}|f(G(t))|=\sup_{t\in[a,b]}|f(t)|=\|f\|_\infty. $$ It remains to be verified that $T$ is linear and a surjection. For the latter, it may be useful to note that $G$ is invertible.

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