Hadamard's Lemma

Hadamard's Lemma is a mathematical statement about functions near a specific point, typically the origin. It states that any smooth function that vanishes at this point can be expressed as the sum of its variables multiplied by other functions. Essentially, if a function equals zero at zero, it can be written as \( f(x) = x_1 g_1(x) + x_2 g_2(x) + \ldots + x_n g_n(x) \). This helps analyze how functions behave close to that point by breaking them down into parts related to each variable, making it easier to study their structure and properties.