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Riesz's lemma state that

If $Y$ is a proper, closed subspace of a normed space $X$, then for any $\epsilon>0$, there exists $x$ in the closed unit ball of $X$ such that $d(x,Y)>1-\epsilon$.

My question is, can we state something stronger--does there exists $x$ in the closed unit ball whose distance to $Y$ equals 1?

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From Diestel's Sequences and Series in Banach Spaces, page 6: "R. C. James (1964) proved that a Banach space $X$ is reflexive if and only if each $x^*\in X^*$ achieves its norm on $B_X$. Using this, one can establish the following: For a Banach space $X$ to have the property that given a proper closed linear subspace $Y$ of $X$ there exists an $x$ of norm-one such that $d(x,Y)\ge 1$ it is necessary and sufficient that $X$ be reflexive." –  David Mitra Feb 6 '13 at 16:24
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A specific counterexample to your claim is: Let $X$ be the subspace of $C[0,1]$ consisting of the members of $C[0,1]$ that vanish at $0$. Let $Y$ be the subspace of $X$ consisting of those functions $f$ for which $\int_0^1 f(x)\,dx=0$. Then there is no $x\in S_X$ with $d(x,Y)\ge1$ (and thus no $x\in B_x$ with $d(x,Y)\ge1$). –  David Mitra Feb 6 '13 at 16:42
    
Thank you very much. I followed your lines and got it. –  Montez Feb 6 '13 at 17:52

1 Answer 1

Answered by David Mitra in the comments:


From Diestel's Sequences and Series in Banach Spaces, page 6: "R. C. James (1964) proved that a Banach space $X$ is reflexive if and only if each $x^*\in X^*$ achieves its norm on $B_X$. Using this, one can establish the following: For a Banach space $X$ to have the property that given a proper closed linear subspace $Y$ of $X$ there exists an $x$ of norm-one such that $d(x,Y)\ge 1$ it is necessary and sufficient that $X$ be reflexive."

A specific counterexample to your claim is: Let $X$ be the subspace of $C[0,1]$ consisting of the members of $C[0,1]$ that vanish at $0$. Let $Y$ be the subspace of $X$ consisting of those functions $f$ for which $\int_0^1 f(x)\,dx=0$. Then there is no $x\in S_X$ with $d(x,Y)\ge 1$ (and thus no $x\in B_X$ with $d(x,Y)\ge 1$).


For completeness: for any $x\in B_X$ we have $$\left|\int_0^1 x(t)\right|\,dt<1 \tag1$$ and the left side of (1) is nothing but $d(x,Y)$.

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