<div class="problem-topics-container">Hilbert spaces, parallelogram law</div>
<problem>
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>H be a Hilbert space and let <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>E be a closed convex subset of <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>H. Prove that there exists a unique element <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">x \in E</annotation></semantics></math>x∈E such that
<div class="math math-display"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo lspace="0em" rspace="0em"></mo><mrow><mo fence="true">∥</mo><mi>x</mi><mo fence="true">∥</mo></mrow><mo lspace="0em" rspace="0em"></mo><mo>=</mo><munder><mrow><mi>inf</mi><mo>⁡</mo></mrow><mrow><mi>y</mi><mo>∈</mo><mi>E</mi></mrow></munder><mtext> </mtext><mo lspace="0em" rspace="0em"></mo><mrow><mo fence="true">∥</mo><mi>y</mi><mo fence="true">∥</mo></mrow><mo lspace="0em" rspace="0em"></mo><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">\norm{x} = \inf_{y \in E}\,\norm{y}.</annotation></semantics></math>∥x∥=y∈Einf​∥y∥.</div>
</problem>
<solution>
<h6>Solution.</h6>
See <a href="/~steven/quals/analysis/10s.7">Fall 2012 - Problem 3</a>.
</solution>