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“Jana Prosason” (Public Administration)( “জনপ্রশাসন”) is authored by Prangovinda Das.(NCBA)
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Overview of Content Rendered on the Cover
- Book Title: Linear Algebra (Multi-Dimensional Vectors)
- Author: Dr. Gunadhar Paria
- Key Mathematical Concept Shown: The proof and formulation of Minkowski’s Inequality for finite sums, which serves as a foundational triangle inequality variant for \(\ell ^{p}\) spaces in advanced linear algebra and mathematical analysis.
Statement of Generalized Minkowski Inequality
The theorem states that if \((a_1, a_2, \dots, a_n)\) and \((b_1, b_2, \dots, b_n)\) are positive real numbers and \(p > 1\), then:
\(\left[\sum _{r=1}^{n}(a_{r}+b_{r})^{p}\right]^{1/p}\le \left[\sum _{r=1}^{n}a_{r}^{p}\right]^{1/p}+\left[\sum _{r=1}^{n}b_{r}^{p}\right]^{1/p}\)
Core Methodology of the Proof
The proof fragment displayed utilizes a strategic algebraic split and applies Hölder’s Inequality (referenced on the page as relation 3 and 5):
- Decomposition: The term \((a_r + b_r)^p\) is written as \((a_r + b_r)(a_r + b_r)^{p-1}\), then expanded linearly:
\(\sum _{r=1}^{n}(a_{r}+b_{r})^{p}=\sum _{r=1}^{n}a_{r}(a_{r}+b_{r})^{p-1}+\sum _{r=1}^{n}b_{r}(a_{r}+b_{r})^{p-1}\) - Applying Hölder’s Inequality: Each of the two split summations is bounded using a conjugate exponent pair \(p\) and \(q\) satisfying \(\frac{1}{p} + \frac{1}{q} = 1\) (or \(p = q(p-1)\)).
- Factoring: The shared sum component \(\left[\sum (a_r + b_r)^p\right]^{1/q}\) is gathered and divided over to the left-hand side, simplifying exponents since \(1 – \frac{1}{q} = \frac{1}{p}\), which yields the final result.
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