Problem 10 Show that (a) For any subset \(S... [FREE SOLUTION]

Chapter 11: Problem 10

Show that (a) For any subset \(S\) of \(V, S \subseteq S^{00}\). (b) If \(S_{1} \subseteq S_{2}\), then \(S_{2}^{0} \subseteq S_{1}^{0}\).

Short Answer

Expert verified

To prove (a) \(S \subseteq S^{00}\), consider any vector \(s \in S\). Since \(s \in S\), for all \(f \in S^0\), we have \(f(s) = 0\). This means \(s\) satisfies the condition for belonging to \(S^{00}\), so \(s \in S^{00}\). Thus, \(S \subseteq S^{00}\). For (b), assume \(S_1 \subseteq S_2\). Given any linear functional \(f \in S_2^0\), we have \(f(s_2) = 0\) for all \(s_2 \in S_2\). Since \(S_1 \subseteq S_2\), for any vector \(s_1 \in S_1\), we have \(f(s_1) = 0\). Hence, \(f \in S_1^0\), and so \(S_2^0 \subseteq S_1^0\).

Step by step solution

Understand the Annihilators

An annihilator is a set of linear functionals that map all vectors in a given set to zero. Specifically, for a subset S of V, the annihilator of S is the set of linear functionals defined by S^0 = {f 鈭� V' : f(s) = 0 for all s 鈭� S}, where V' is the dual space of V. This means that S^0 contains all the linear functionals that annihilate S. Similarly, S^{00} is the annihilator of S^0, sometimes called the second annihilator. It consists of all vectors s 鈭� V such that f(s) = 0 for all f 鈭� S^0.

Prove (a) That S is a Subset of S^{00}

To prove that S 鈯� S^{00}, we need to show that for any vector s 鈭� S, it is also true that s 鈭� S^{00}. Let's consider any arbitrary vector s 鈭� S: 1. Since s 鈭� S, by the definition of the annihilator, for all f 鈭� S^0, we have f(s) = 0. 2. Now take an arbitrary element f 鈭� S^0, since f(s) = 0 for all s 鈭� S, s satisfies the condition for belonging to S^{00}. 3. Thus, s 鈭� S^{00}. Since this is true for any arbitrary element s 鈭� S, we can conclude that S 鈯� S^{00}.

Prove (b) If S鈧� 鈯� S鈧�, Then S鈧俕0 鈯� S鈧乛0

To prove that if S鈧� 鈯� S鈧�, then S鈧俕0 鈯� S鈧乛0, we need to show that for any linear functional f 鈭� S鈧俕0, it is also true that f 鈭� S鈧乛0. Let's consider any arbitrary linear functional f 鈭� S鈧俕0. 1. Since f 鈭� S鈧俕0, by the definition of the annihilator, for all vectors s鈧� 鈭� S鈧�, we have f(s鈧�) = 0. 2. Now S鈧� 鈯� S鈧�, so any vector s鈧� 鈭� S鈧� must also be an element of S鈧� (i.e., s鈧� 鈭� S鈧�). 3. Thus, for all s鈧� 鈭� S鈧�, f(s鈧�) = 0, since f(s鈧�) = 0 for all s鈧� 鈭� S鈧�. 4. This satisfies the condition for f belonging to S鈧乛0, meaning f 鈭� S鈧乛0. Since this is true for any arbitrary element f 鈭� S鈧俕0, we can conclude that S鈧俕0 鈯� S鈧乛0.

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91影视

Show that (a) For any subset \(S\) of \(V, S \subseteq S^{00}\). (b) If \(S_{1} \subseteq S_{2}\), then \(S_{2}^{0} \subseteq S_{1}^{0}\).

Short Answer

Step by step solution

Understand the Annihilators

Prove (a) That S is a Subset of S^{00}

Prove (b) If S鈧� 鈯� S鈧�, Then S鈧俕0 鈯� S鈧乛0

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