Weakly

Weakly is an adverb formed from the adjective weak. It indicates that an action is performed with little strength, energy, or force. In everyday language it can describe how something is done or felt, such as a weakly expressed remark or a weakly audible sound. It is also used in scientific contexts to describe phenomena of small magnitude or intensity.

In mathematics and logic, weakly is a technical qualifier used with convergence, order, and topology. For a sequence (x_n) in a normed space, x_n converges weakly to x if for every continuous linear functional f, f(x_n) → f(x). This notion is weaker than convergence in norm; norm convergence implies weak convergence, but not the other way around. In finite-dimensional spaces, weak convergence and norm convergence coincide. In Hilbert spaces the condition can be stated as <x_n, y> → <x, y> for every y in the space. Related ideas include the weak topology and the weak- topology on dual spaces.

Weakly is also used to describe order relations in sequences and functions, such as weakly increasing or weakly decreasing, meaning non-decreasing or non-increasing without requiring strict monotonicity.

Beyond mathematics, weakly appears in physics and probability to denote phenomena of small or nonstrong interactions, influences, or convergences. In probability theory, weak convergence, or convergence in distribution, describes convergence of probability measures in a manner weaker than pointwise convergence.

Overall, weakly serves as a precise term in specialized contexts while retaining a general meaning of low strength or intensity in everyday language.