References
[1] Xia Daoxing et al. Real variable function theory and functional analysis (Volume 2, revised second edition). Beijing: Higher Education Press, 2010
Exercise Let represent the entire sequence that converges to zero. According to linear operations and norms
The resulting Banach space. Proof: .
Proof: Record the number as , and the rest are all sequences, that is
Then. Note that since the linear subspace formed by is dense in , it is determined by. For, note
Take, due to
Therefore, it is a linear mapping to. On the contrary, for, at that time, due to
Therefore, it is the linear norm-preserving isomorphism of .
Machine-translated from the Chinese original 一道泛函分析的课后作业题 (published 2024-05-17). WeChat original ↗