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Riesz's lemma (after Frigyes Riesz) is a lemma in functional analysis. It specifies (often easy to check) conditions that guarantee that a subspace in a normed linear space is dense. The lemma may also be called the Riesz lemma or Riesz inequality. It can be seen as a substitute for orthogonality when one is not in an inner product space. == The result == Before stating the result, we fix some notation. Let ''X'' be a normed linear space with norm |·| and ''x'' be an element of ''X''. Let ''Y'' be a closed subspace in ''X''. The distance between an element ''x'' and ''Y'' is defined by : Now we can state the Lemma: Riesz's Lemma. Let ''X'' be a normed linear space, ''Y'' be a closed proper subspace of ''X'' and α be a real number with Then there exists an ''x'' in ''X'' with |''x''| = 1 such that |''x'' − ''y''| > α for all ''y'' in ''Y''. ''Remark 1.'' For the finite-dimensional case, equality can be achieved. In other words, there exists ''x'' of unit norm such that ''d''(''x'', ''Y'') = 1. When dimension of ''X'' is finite, the unit ball ''B'' ⊂ ''X'' is compact. Also, the distance function ''d''(· , ''Y'') is continuous. Therefore its image on the unit ball ''B'' must be a compact subset of the real line, proving the claim. ''Remark 2.'' The space ℓ∞ of all bounded sequences shows that the lemma does not hold for α = 1. The proof can be found in functional analysis texts such as Kreyszig. An (online proof from Prof. Paul Garrett ) is available. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Riesz's lemma」の詳細全文を読む スポンサード リンク
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