A function belonging to L2 and L_infinity means that it satisfies the following conditions: it is square integrable, meaning the integral of its squared absolute value is finite, and it is bounded, meaning its absolute value is always less than or equal to a constant.
Yes, a function can belong to L2 and L_infinity but not be vanishing. This means that the function can have non-zero values but still satisfy the conditions of being square integrable and bounded.
A function can belong to L2 and L_infinity but not be vanishing if it has finite support, meaning it is non-zero only on a finite interval. In this case, the integral of its squared absolute value over the entire domain will still be finite, and it will be bounded since it is only non-zero on a finite interval.
Yes, there are many real-life examples of such functions. For instance, the function representing the size of a population over time can belong to L2 and L_infinity if it is bounded and the population does not decrease to zero. Another example is a function representing the temperature of a system over time, which can belong to L2 and L_infinity if it is bounded and the temperature does not reach absolute zero.
A function belonging to L2 and L_infinity but not vanishing has important implications in mathematical analysis and signal processing. It allows for the use of powerful mathematical tools, such as Fourier analysis and Hilbert spaces, to study and manipulate the function. This is because L2 and L_infinity are important function spaces that are commonly used in these fields.