习题1.1
第 1 题
证明: \(n\) 维 Euclid 空间中的距离 \(\left\lVert\mathbf{X} - \mathbf{Y}\right\rVert _n\) 满足正定性、对称性与三角不等式.
证明
不妨设 \(\mathbf{X} = (x_1, x_2, \cdots, x_n), \mathbf{Y} = (y_1, y_2, \cdots, y_n)\), 则 \(\left\lVert\mathbf{X} - \mathbf{Y}\right\rVert _n = \sqrt{\sum_{i = 1}^n(x_i - y_i)^2}\).
(1) 正定性: 显然 \(\left\lVert\mathbf{X} - \mathbf{Y}\right\rVert _n = \sqrt{\sum_{i = 1}^n(x_i - y_i)^2} \ge 0\), 且 \(\left\lVert\mathbf{X} - \mathbf{Y}\right\rVert _n = 0\) 当且仅当 \(\forall i \in [1, n]\) 均有 \(x_i - y_i = 0\), 此即 \(\mathbf{X} = \mathbf{Y}\).
(2) 对称性: 显然 \(\left\lVert\mathbf{Y} - \mathbf{X}\right\rVert _n = \sqrt{\sum_{i = 1}^n(x_i - y_i)^2} = \left\lVert\mathbf{X} - \mathbf{Y}\right\rVert _n\)
(3) 三角不等式: 再设 \(\mathbf{Z} = (z_1, z_2, \cdots, z_n)\). 要证 \(\left\lVert\mathbf{X} - \mathbf{Y}\right\rVert _n \le \left\lVert\mathbf{X} - \mathbf{Z}\right\rVert _n + \left\lVert\mathbf{Z} - \mathbf{Y}\right\rVert _n\), 只需证 \(\sqrt{\sum_{i = 1}^n(x_i - y_i)^2} \le \sqrt{\sum_{i = 1}^n(x_i - z_i)^2} + \sqrt{\sum_{i = 1}^n(z_i - y_i)^2}\). 由于
其中不等号由柯西不等式得到. 故三角不等式得证.
第 2 题
求下列集合 \(\Omega\) 的内部、外部、边界和闭包.
(1) \(\Omega\) 为 \(\mathbb{R}^2\) 的子集, \(\Omega = \{(x, y) | x^2 + y^2 = 1\}\);
(2) \(\Omega\) 为 \(\mathbb{R}^3\) 的子集, \(\Omega = \{(x, y, z) | 1 \le x^2 + y^2 + z^2 < 4\}\).
解
(1) 内部: \(\varnothing\), 外部: \(\{(x, y) | x^2 + y^2 \neq 1\}\), 边界: \(\{(x, y) | x^2 + y^2 = 1\}\), 闭包: \(\{(x, y) | x^2 + y^2 = 1\}\).
(2) 内部: \(\{(x, y, z) | 1 < x^2 + y^2 + z^2 < 4\}\), 外部 \(\{(x, y, z) | x^2 + y^2 + z^2 < 1 \lor x^2 + y^2 + z^2 > 4\}\), 边界: \(\{(x, y, z) | x^2 + y^2 + z^2 = 1 \lor x^2 + y^2 + z^2 = 4\}\), 闭包: \(\{(x, y, z) | 1 \le x^2 + y^2 + z^2 \le 4\}\).
第 3 题
证明下列命题:
(1) 已知 \(S \subset \mathbb{R}^n\), 则 \(S\) 为开集 \(\iff\) \(S = \mathring{S}\);
(2) 若 \(S \subset \mathbb{R}^n\) 为开集, 则 \(S \cap \partial S = \varnothing\);
(3) 任意多个开集之并为开集; 有限个开集之交为开集;
(4) 若 \(A, B \subset \mathbb{R}^n\), 记 \(S = A \cap B, T = A \cup B\), 则 \(\mathring{S} = \mathring{A} \cap \mathring{B}, \mathring{T} \supset \mathring{A} \cup \mathring{B}\);
(5) 若 \(A \subset \mathbb{R}^n\), 则集合 \(\mathring{A}\) 的内部等于 \(\mathring{A}\).
第 4 题
证明下列命题:
(1) 已知 \(S \subset \mathbb{R}^n\), 则 \(S\) 为闭集 \(\iff\) \(S = \overline{S} \iff \partial S \subset S\);
(2) 若 \(A, B \subset \mathbb{R}^n\), 则 \(\overline{A}\backslash\overline{B} \subset \overline{A \backslash B}, \overline{A}\cup\overline{B} \subset \overline{A\cup B}\) (事实上他们相等), \(\overline{A}\cap\overline{B} \supset \overline{A\cap B}\);
(3) 若 \(P_1, P_2, \cdots, P_k \in \mathbb{R}^n\), 则 \(\{P_1, P_2, \cdots, P_k\}\) 为闭集;
(4) 任意多个闭集之交为闭集; 有限个闭集之并为闭集.
证明
第 5 题
证明下列命题:
(1) 已知 \(S \subset \mathbb{R}^n\), 则 \(\overset{\circ}{S}\) 等于 \(S\) 的余集的闭包的余集; \(\overline{S}\) 等于 \(S\) 的余集的内部的余集;
(2) 若 \(A \subset \mathbb{R}^n\), 则 \(\overline{A} = A \cup \partial A = \overset{\circ}{A} \cup \partial A, \overset{\circ}{A} = A \backslash\partial A = \overline{A}\backslash\partial A, \partial A = \partial(\mathbb{R}^n\backslash A)\);
(3) 若 \(A \subset \mathbb{R}^n\), 则 \(\partial(\overset{\circ}{A}), \partial(\overline{A}) \subset \partial A\);
(4) 若 \(A, B \subset \mathbb{R}^n\), 则 \(\partial(A\cup B) \subset \partial A \cup \partial B\);
(5) \(\partial A = \varnothing\) \(\iff\) \(A\) 既是开集又是闭集.
证明
第 6 题
证明: \(\mathbb{R}^n\) 中的点列 \(\{\mathbf{X}_k\}\) 为 Cauchy 列当且仅当 \(\{\mathbf{X}_k\}\) 的 \(n\) 个分量构成的 \(n\) 个实数列 \(\{x_k^{(i)}\}_{k=1}^{+\infty}(i = 1, 2, \cdots, n)\) 均为 Cauchy 列.
证明
第 7 题
下列集合中, 哪些是连通的, 哪些是非连通的?
(1) \(D = \{(x, y) | y \neq 0\}\);
(2) \(D = \{(x, y) | 0 < x^2 + y^2 \le 2\}\);
(3) \(\Omega = \{(x, y, z) | x^2 + y^2 \neq 0\}\);
(4) \(\Omega = \{(x, y, z) | 1 < x^2 + y^2 + z^2 \le 4\}\).
解
(1) 不是.
(2) 是.
(3) 是.
(4) 是.
第 8 题
连通的闭集是否为闭区域? 如果是,请证明; 如果不是, 请举出反例.
解
第 9 题
证明: \(\mathbb{R}^n\) 中的收敛点列必为有界点列.
证明