- #1
Chris L T521
Gold Member
MHB
- 915
- 0
Here's this week's problem.
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Background Info: Let $X$ be a normed linear space. The linear operator $J:X\rightarrow X^{\ast\ast}$ defined by
\[J(x)[\psi] = \psi(x) \text{ for all $x\in X$, $\psi\in X^{\ast}$}\]
is called the natural embedding of $X$ into $X^{\ast\ast}$.
Problem: Let $X$ be a normed linear space. Show that the natural embedding $J:X\rightarrow X^{\ast\ast}$ is an isometry.
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Background Info: Let $X$ be a normed linear space. The linear operator $J:X\rightarrow X^{\ast\ast}$ defined by
\[J(x)[\psi] = \psi(x) \text{ for all $x\in X$, $\psi\in X^{\ast}$}\]
is called the natural embedding of $X$ into $X^{\ast\ast}$.
Problem: Let $X$ be a normed linear space. Show that the natural embedding $J:X\rightarrow X^{\ast\ast}$ is an isometry.
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