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However, if a sequence converges in and in , then its limit must be the same in both spaces.For 1" /, the dual vector space to is given by integrating against functions in , where .Explicitly, the vector sum and the scalar action for infinite sequences of real (or complex) numbers are given by: space is obtained—as seen below — by considering vectors, not only with finitely or countably-infinitely many components, but with "arbitrarily many components"; in other words, functions.An integral instead of a sum is used to define the are sometimes called quadratically integrable functions, square-integrable functions or square-summable functions, but sometimes these terms are reserved for functions that are square-integrable in some other sense, such as in the sense of a Riemann integral (Titchmarsh 1976).An analogy to this is suggested by taxi drivers in a grid street plan who should measure distance not in terms of the length of the straight line to their destination, but in terms of the rectilinear distance, which takes into account that streets are either orthogonal or parallel to each other.

In many situations, the Euclidean distance is insufficient for capturing the actual distances in a given space.While the use of functions is not as common as , they are very important in analysis and partial differential equations.spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces.Here, the weather never puts a damper on your plans.Invite your friends to design and decorate your residence with you, or buy a pre-made home and have a housewarming party.