10. Let V = Mfin (N, R), the real space of all maps f from N to R such that_spt(f) = {i € N[ƒ(i) ‡ 0} is finite. Define (, ): V x V → R by (f, g) = Σï₁ f(i)g(i). Prove that (, ) is an inner product space on V.
10. Let V = Mfin (N, R), the real space of all maps f from N to R such that_spt(f) = {i € N[ƒ(i) ‡ 0} is finite. Define (, ): V x V → R by (f, g) = Σï₁ f(i)g(i). Prove that (, ) is an inner product space on V.
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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![### Problem 10
Let \( V = M_{\text{fin}}(\mathbb{N}, \mathbb{R}) \), the real space of all maps \( f \) from \( \mathbb{N} \) to \( \mathbb{R} \) such that
\[ \text{spt}(f) = \{ i \in \mathbb{N} | f(i) \neq 0 \} \]
is finite. Define \(\langle \cdot , \cdot \rangle : V \times V \to \mathbb{R}\) by
\[ \langle f, g \rangle = \sum_{i=1}^{\infty} f(i)g(i). \]
Prove that \(\langle \cdot , \cdot \rangle \) is an inner product space on \( V \).
### Problem 11
Let \((V, \langle \cdot, \cdot \rangle )\) be a complex inner product space. For vectors \( v, w \), set
\[ \langle v, w \rangle _{\mathbb{R}} = \frac{1}{2} \left( \langle v, w \rangle + \langle v, v \rangle \right) \]
Consider \( V \) to be a real vector space. Rename \(\langle \cdot, \cdot \rangle _{\mathbb{R}} \) as \(\langle \cdot, \cdot \rangle \). Prove that \(\langle \cdot, \cdot \rangle \) is an inner product on \( V \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fae19e8bd-5915-473c-8a03-9ac053dffb7e%2Fd273203f-ea5e-4057-a853-6919a4db5035%2Fzpwb57m_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem 10
Let \( V = M_{\text{fin}}(\mathbb{N}, \mathbb{R}) \), the real space of all maps \( f \) from \( \mathbb{N} \) to \( \mathbb{R} \) such that
\[ \text{spt}(f) = \{ i \in \mathbb{N} | f(i) \neq 0 \} \]
is finite. Define \(\langle \cdot , \cdot \rangle : V \times V \to \mathbb{R}\) by
\[ \langle f, g \rangle = \sum_{i=1}^{\infty} f(i)g(i). \]
Prove that \(\langle \cdot , \cdot \rangle \) is an inner product space on \( V \).
### Problem 11
Let \((V, \langle \cdot, \cdot \rangle )\) be a complex inner product space. For vectors \( v, w \), set
\[ \langle v, w \rangle _{\mathbb{R}} = \frac{1}{2} \left( \langle v, w \rangle + \langle v, v \rangle \right) \]
Consider \( V \) to be a real vector space. Rename \(\langle \cdot, \cdot \rangle _{\mathbb{R}} \) as \(\langle \cdot, \cdot \rangle \). Prove that \(\langle \cdot, \cdot \rangle \) is an inner product on \( V \).
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