Norm on Vector Space is Continuous Function
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Theorem
Let $V$ be a vector space with norm $\norm {\, \cdot \,}$.
The function $\norm {\, \cdot \,}: V \to \R$ is continuous.
Proof
Let $x_n \to x$ in $V$.
We have:
- $x_n \to x \implies \norm {x_n - x} \to 0$
By the Reverse Triangle Inequality:
- $\size {\norm {x_n} - \norm x} \le \norm {x_n - x}$
Hence:
- $\size {\norm {x_n} - \norm {x_n} } \to 0$
Should the above read $\size {\norm {x_n} - \norm x} \to 0$?
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Thus:
- $\norm {x_n} \to \norm x$
Explain why, with use of links, this shows continuity.
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