Notation of Holder space

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The question is from book Abel Integral Equation, Sergio and Rudolf.
From section 4.2.1 they denoted Holder space as $C^{\alpha}[0,a]$. I know holder space as $C^{k,\alpha}[0,a]$. Are both same? or is it only space of holder continuous functions with exponent $\alpha$ that is $C^{0,\alpha}[0,a]$? I am not able to understand this notation exactly.enter image description here