In Gelfand and Formin Calculus of Variation, the increment of the functional is
$$ \Delta J [h] = J[y+h] - J[y]$$
it is supposed that the increment can also be written in the form of
$$\Delta J[h] = \varphi [h] + \varepsilon ||h||$$
however I do not see how the 2nd definition of the functional increment picks up a $ \varepsilon ||h||$ term. What is the significance of the $\varepsilon ||h||$ term in the 2nd definition.
It is confusing what does the $\varepsilon$ multiply $||h||$ mean in the context of a functional
Hope I could get this confusion clarified, as it was not explained the reason why this supposition was made.