Word Count : 608

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Let's analyze each statement carefully. ### (a) The function \( \|x\| \) defined on \( \mathbb{R}^n \) as \[ \|x\|=\sum_{j=1}^{n}|a_j| \quad \text{for } x=(a_1,a_2,\dots,a_n)\in\mathbb{R}^n \] #### Solution: This is known as the \( \ell_1 \)-norm or Manhattan norm in \( \mathbb{R}^n \). The function satisfies the three properties of a norm: 1. Non-negativity: \( \|x\| \geq 0 \), and \( \|x\| = 0 \) if and only if \( x = 0 \). 2. Scalar multiplication: \( \|\alpha x\| = |\alpha| \|x\| \) for any scalar \( \alpha \). 3. Triangle inequality: \( \|x + y\| \leq \|x\| + \|y\| \). Since these conditions hold, this function is indeed a valid norm. Verdict: True ✅ --- ### (b) \( C_0 \) is a Banach space. #### Solution: The space \( C_0 \) consists of all sequences of real or complex numbers that converge to zero at infinity, equipped with the supremum norm: ________ _______ ____ __________ ____ ___ ______ ___.
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