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Let $X$ be an inner product space, and $A$ a closed subspace of $X$ with $A\neq X$.

Provide an example of $X$ and $ A$ where $\inf_{y\in A} \Vert x-y \Vert <1$ for every $x \in X $ with $\Vert x \Vert =1$

My attempt:

I try to start with $A=\{ f\in C[0,1]: \int_{0}^{1}fg=0\}$, but I am stuck here, I don't know how to use the incompleteness.

Any help would be appreciated.

Conifold
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Flashhh
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  • Take $A=X$, then $\inf=0$ for all $x$. – Conifold Dec 17 '19 at 05:24
  • @Conifold Sorry, I edited $A \neq X$ – Flashhh Dec 17 '19 at 05:24
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    Then take as $A$ orthogonal complement to some discontinuous function $f$ in $L^2([0,1])$ intersected with $C([0,1])$. We can normalize to have $|f|=1$. Since $\pm f$ is not in $C([0,1])$ anything there with $|x|=1$ will be at a distance $<1$ from $A$ by the Pythagorean theorem. This requires your inner product space to be incomplete. If it is complete this is impossible. – Conifold Dec 17 '19 at 05:34
  • Ries’s Lemma provides a construction of the object you are seeking for a general Banach space. Also, notice that if Y is closed, then the infimum is the generalization of orthogonality for Banach spaces. – gdepaul Dec 17 '19 at 05:57
  • @gdepaul I don't see how Riesz's lemma produces an answer to this question – Ben Grossmann Dec 17 '19 at 07:34
  • Following up Conifold's comment: a convenient $f$ to work with would be a step-function such as $$ f(x) = \begin{cases} 0 & x < \frac 12\ 1 & x \geq \frac 12 \end{cases} $$ – Ben Grossmann Dec 17 '19 at 07:37

1 Answers1

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See the discussion in the comments. Take $X = C[0,1]$ with the $\int fg$ inner-product. One example of such a space $A$ would be $$ A = \left\{f \in C[0,1]:\int_0^{1/2}f(x)\,dx = 0\right\}. $$

Ben Grossmann
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