Continuous Linear Functionals
Why
We can characterize the continuous linear functionals.
Main result
Let $F$ be a linear functional on a normed
space $(V, \norm{\cdot })$. The following are
equivalent:
For this reason we often call
continuous linear
functionals the bounded
linear functionals or call them
continuous bounded linear
functionals.- $F$ is continuous;
- $F$ is continuous at 0;
- $\sup_{\norm{x} \leq 1}\set{\abs{F(x)}} < \infty$.1
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