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Module 2 : Fundamentals of Vector Spaces

Section 4 : Inner Product Spaces and Hilbert Spaces

Axioms 2 and 3 imply that the inner product is linear in the first entry. The quantity is a candidate function for defining norm on the inner product space Axioms 1 and 3 imply that and axiom 4 implies that for If we show that satisfies triangle inequality, then defines a norm on space . We first prove Cauchy-Schwarz inequality, which is generalization of equation (cos), and proceed to show that defines the well known 2-norm on i.e. . Lemma 41 (Cauchey- Schwarz Inequality): Let denote an inner product space. For all ,the following inequality holds
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The equality holds if and only if $\QTR{bf}{x}=$ $\lambda\QTR{bf}{y}$ or Proof: If , the equality holds trivially so we assume Then, for all scalars $\lambda,$ we have
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In particular, if we choose then, using axiom 1 in the definition of inner product, we have
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The triangle inequality can be can be established easily using the Cauchy-Schwarz inequality as follows
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Thus, the candidate function satisfies all the properties necessary to define a norm, i.e.
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Thus, the function indeed defines a norm on the inner product space In fact the inner product defines the well known 2-norm on i.e.
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and the triangle inequality can be stated as
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