Lesson plan
C space
In the mathematical field of functional analysis, the space denoted by c is the vector space of all convergent sequences \left(x_n\right) of real numbers or complex numbers. When equipped with the uniform norm: \|x\|_\infty = \sup_n |x_n| the space c becomes a Banach space. It is a closed linear subspace of the space of bounded sequences, \ell^\infty , and contains as a closed subspace the Banach space c_0 of sequences converging to zero. The dual of c is isometrically isomorphic to \ell^1, as is that of c_0. In particular, neither c nor c_0 is reflexive. In the first case, the isomorphism of \ell^1 with c^* is given as follows. If \left(x_0, x_1, \ldots\right) \in \ell^1, then the pairing with an element \left(y_0, y_1, \ldots\right) in c is given by x_0\lim_{n\to\infty} y_n + \sum_{i=0}^\infty x_{i+1} y_i. This is the Riesz representation theorem on the ordinal \omega . For c_0, the pairing between \left(x_i\right) in \ell^1 and \left(y_i\right) in c_0 is given by \sum_{i=0}^\infty x_iy_i.
Article summary
In the mathematical field of functional analysis, the space denoted by c is the vector space of all convergent sequences \left(x_n\right) of real numbers or complex numbers. When equipped with the uniform norm: \|x\|_\infty = \sup_n |x_n| the space c becomes a Banach space. It is a closed linear subspace of the space of bounded sequences, \ell^\infty , and contains as a closed subspace the Banach space c_0 of sequences converging to zero. The dual of c is isometrically isomorphic to \ell^1, as is that of c_0. In particular, neither c nor c_0 is reflexive. In the first case, the isomorphism of \ell^1 with c^* is given as follows. If \left(x_0, x_1, \ldots\right) \in \ell^1, then the pairing with an element \left(y_0, y_1, \ldots\right) in c is given by x_0\lim_{n\to\infty} y_n + \sum_{i=0}^\infty x_{i+1} y_i. This is the Riesz representation theorem on the ordinal \omega . For c_0, the pairing between \left(x_i\right) in \ell^1 and \left(y_i\right) in c_0 is given by \sum_{i=0}^\infty x_iy_i.
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Discussion questions
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