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THEOREM 9.2. If X is an inner product space, the inner product (x, y) is a con- tinuous function mapping Xx X into F. Proof. Consider the fixed point in the range (x2, y2). Now let X3 = X1 X₂ and |(x₁, y₁) (X2, Y₂)] = [(X2 + X3, Y2 + Y3) — (x₂, Y₂)]. which implies Y3 = V₁ V2,

THEOREM 9.2. If X is an inner product space, the inner product (x, y) is a con- tinuous function mapping Xx X into F. Proof. Consider the fixed point in the range (x2, y2). Now let X3 = X1 X₂ and |(x₁, y₁) (X2, Y₂)] = [(X2 + X3, Y2 + Y3) — (x₂, Y₂)]. which implies Y3 = V₁ V2,

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter5: Rings, Integral Domains, And Fields
Section5.4: Ordered Integral Domains
Problem 1E: Complete the proof of Theorem 5.30 by providing the following statements, where and are arbitrary...
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Request explain how continuity is mapping is evident

THEOREM 9.2. If X is an inner product space, the inner product (x, y) is a con- tinuous function mapping Xx X into F. Proof. Consider the fixed point in the range (x2, 1₂). Now let and which implies X3 = x1 - x₂ Y3 = V₁ - Y2, (x₁, y₁) (x2, Y2)] = [(X₂ + X3, Y2 + Y3) − (X2, Y2)]. Expanding the first inner product and appealing to the Cauchy-Schwarz inequality, noted in Theorem 1.1, we have |(X2, Y3) + (X3, Y₂) + (X3, Y3)| ≤ ||X2|| ||Y1 — Y2|| + ||Y2|| ||X₁ — X₂|| and the continuity of the above mapping is evident. + ||X₁ X₂||||y₁ - y2||,
Transcribed Image Text:THEOREM 9.2. If X is an inner product space, the inner product (x, y) is a con- tinuous function mapping Xx X into F. Proof. Consider the fixed point in the range (x2, 1₂). Now let and which implies X3 = x1 - x₂ Y3 = V₁ - Y2, (x₁, y₁) (x2, Y2)] = [(X₂ + X3, Y2 + Y3) − (X2, Y2)]. Expanding the first inner product and appealing to the Cauchy-Schwarz inequality, noted in Theorem 1.1, we have |(X2, Y3) + (X3, Y₂) + (X3, Y3)| ≤ ||X2|| ||Y1 — Y2|| + ||Y2|| ||X₁ — X₂|| and the continuity of the above mapping is evident. + ||X₁ X₂||||y₁ - y2||,
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