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,
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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||,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe15b7304-cc73-4505-92c3-23aa2fda4f71%2Fc2daffc8-df32-4324-adbe-f8131fbb5cb0%2F9zymyef_processed.png&w=3840&q=75)
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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