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Spaţiu vectorial normat

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Fie \mathcal X un spaţiu vectorial peste corpul \mathcal K (\mathcal \Gamma sau \mathbb C). Se numeşte normă pe \mathcal X orice aplicaţie \| \|: \mathcal X \rightarrow \mathbb R cu proprietăţile:

  • \|x \| \ge 0, \; \forall x \in \mathcal X şi \| x\|=0 dacă şi numai dacă x= 0_{\mathcal X}.
  •  \|\lambda x \| = |\lambda | \cdot \|x\| , \; \forall x \in \mathcal X.
  • \|x+y\| \le \|x\| + \|y\|, \; \forall x, y \in \mathcal X.

Perechea (\mathcal X, \| \|) se numeşte spaţiu normat.

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