STnormejas

STnormejas are a class of time-dependent norm-like functionals defined on vector spaces, used in theoretical analysis to measure the size of elements as a function of time or along trajectories. The term combines a temporal aspect with a norm-like concept, emphasizing that the measured magnitude can vary with time while preserving a norm structure at each fixed time.

Definition and basic properties: For each nonnegative time t, the map x ↦ ||x||_ST(t) is a norm on the underlying vector space, and the family t ↦ ||x||_ST(t) may exhibit various regularity properties such as measurability or continuity. In many formulations, the time dependence is chosen to be continuous or piecewise smooth, and additional conditions (such as monotonicity in t for fixed x) may be imposed to suit a given application. When t is fixed, STnormejas reduce to ordinary norms, while the time dependence encodes dynamics or evolving sensitivity.

Common instantiations: A simple realization is a time-weighted norm, ||x||_ST(t) = w(t) ||x||, where w(t) ≥ 0. A more dynamic version uses a path x: [0,T] → V and defines ||x||_ST = sup_{s ≤ t} κ(t−s) ||x(s)|| with a kernel κ that may impose decay or emphasis over history. Both forms allow the same underlying vector space to be evaluated in a time-aware way, while preserving the standard norm properties at each moment.

Applications and context: STnormejas appear in time-series analysis, control theory, and signal processing to quantify evolving magnitudes, stability under non-autonomous dynamics, and time-dependent regularization in optimization. They provide a flexible framework for integrating temporal behavior into norm-based analysis, while remaining consistent with classical norms when the time component is constant.

See also: norm, time-dependent norms, quasi-norm, seminorm, Banach space, Lyapunov functions.