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<p>We study the norming points and norming functionals of symmetric operators on <mml:math alttext="$L_p$"><mml:mrow><mml:msub><mml:mi>L</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:mrow></mml:math> spaces for <mml:math alttext="$p=2m$"><mml:mrow><mml:mi>p</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mi>m</mml:mi></mml:mrow></mml:math> or <mml:math alttext="$p={2m}/({2m-1})$"><mml:mrow><mml:mi>p</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>m</mml:mi></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:mi>m</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:math>. We prove some general result relating uniqueness of minimal projection to the set of norming functionals. As a main application, we obtain that the Fourier projection onto span <mml:math alttext="$[1,sin x,cos x]$"><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>sin</mml:mi><mml:mo>&#x2061;</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>cos</mml:mi><mml:mo>&#x2061;</mml:mo><mml:mi>x</mml:mi></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:math> is a unique minimal projection in <mml:math alttext="$L_p$"><mml:mrow><mml:msub><mml:mi>L</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:mrow></mml:math>.</p

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