Normed real space X, f in X*, ||f|| = 1, sigma in (0, 1), C:= {x in X : f(x) >= ...
Let X be a real normed space, \(\displaystyle f\, \in\, X*,\, \|\, f \,\| \, =\, 1,\) and let \(\displaystyle \sigma\, \in \, (0,\, 1).\)
\(\displaystyle C\, :=\, \left\{x\, \in\, X\, :\, f(x)\, \geq\, \sigma\, \|\,x\, \| \right\}\)
Prove that:
\(\displaystyle C^{\circ}\, =\, \left\{x\, \in\, X\, :\, f(x)\, >\, \sigma\,\|\,x\, \|\right\}\)
Can you help me with this one?
Let X be a real normed space, \(\displaystyle f\, \in\, X*,\, \|\, f \,\| \, =\, 1,\) and let \(\displaystyle \sigma\, \in \, (0,\, 1).\)
\(\displaystyle C\, :=\, \left\{x\, \in\, X\, :\, f(x)\, \geq\, \sigma\, \|\,x\, \| \right\}\)
Prove that:
\(\displaystyle C^{\circ}\, =\, \left\{x\, \in\, X\, :\, f(x)\, >\, \sigma\,\|\,x\, \|\right\}\)
Can you help me with this one?
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