7. Assume that Vi, i = 1,2 are vector spaces over F E {R, C} and (,), i = 1, 2 is an inner product on V₂. Set V = V₁ V₂ and define (, ): V x V → F by ((U₁, U2), (v₁, v₂)) = (U₁, v₁)₁ + (u2; v₂) 2 for u₁, v₁ € V₁, U2, v2 € V₂. Determine whether (, ) is an inner product on V. Prove your conclusion.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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7. Assume that Vi, i = 1, 2 are vector spaces over F E {R, C) and (, ),, i = 1, 2
is an inner product on V₂. Set V = V₁ V₂ and define (, ): V x V → F by
((u₁, U₂), (v₁, v₂)) = (u₁, v₁)₁ + (u2; v2) 2
1
for u₁, v₁ € V₁, U2, v2 € V₂. Determine whether (, ) is an inner product on
V. Prove your conclusion.
8. Let (V, (, )) be an inner product space and L = (v₁, V2,..., Vn.) a sequence
Transcribed Image Text:7. Assume that Vi, i = 1, 2 are vector spaces over F E {R, C) and (, ),, i = 1, 2 is an inner product on V₂. Set V = V₁ V₂ and define (, ): V x V → F by ((u₁, U₂), (v₁, v₂)) = (u₁, v₁)₁ + (u2; v2) 2 1 for u₁, v₁ € V₁, U2, v2 € V₂. Determine whether (, ) is an inner product on V. Prove your conclusion. 8. Let (V, (, )) be an inner product space and L = (v₁, V2,..., Vn.) a sequence
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