Question

Let $f: X \to Y$ be a function between metric spaces $(X, d)$ and $(Y, \rho)$, and let $K \subseteq X$. (4.1) Let $x_0 \in X$. What does it mean to say that $f$ is continuous at the point $x_0$. (4.2) Prove that the following statements are equivalent. (a) $f$ is continuous. (b) For every subset $A \subseteq X$, $f(\overline{A}) \subseteq \overline{f(A)}$. (c) If $B$ is closed in $Y$, then $f^{-1}(B)$ is closed in $X$. (d) If $D$ is open in $Y$, then $f^{-1}(D)$ is open in $X$. (4.3) Define what $K$ is a compact subset of $X$ means and explain any terminology that you use in your definition. (4.4) If $f$ is continuous and $K$ is compact, use the definition of compactness to show that $f(K)$ is a compact subset of $Y$. (4.5) Suppose that $X$ is a vector space over the field of complex numbers $\mathbb{C}$ with inner product $(\cdot, \cdot)$. Define $||\cdot||: X \to \mathbb{R}$ by $||x|| = \sqrt{(x, x)}$. Use the Cauchy-Bunyakowski-Schwarz (CBS) inequality to show that $||\cdot||$ is a norm on $X$.

          Let $f: X \to Y$ be a function between metric spaces $(X, d)$ and $(Y, \rho)$, and let $K \subseteq X$.
(4.1) Let $x_0 \in X$. What does it mean to say that $f$ is continuous at the point $x_0$.
(4.2) Prove that the following statements are equivalent.
(a) $f$ is continuous.
(b) For every subset $A \subseteq X$, $f(\overline{A}) \subseteq \overline{f(A)}$.
(c) If $B$ is closed in $Y$, then $f^{-1}(B)$ is closed in $X$.
(d) If $D$ is open in $Y$, then $f^{-1}(D)$ is open in $X$.
(4.3) Define what $K$ is a compact subset of $X$ means and explain any terminology that you use in your definition.
(4.4) If $f$ is continuous and $K$ is compact, use the definition of compactness to show that $f(K)$ is a compact subset of $Y$.
(4.5) Suppose that $X$ is a vector space over the field of complex numbers $\mathbb{C}$ with inner product $(\cdot, \cdot)$.
Define $||\cdot||: X \to \mathbb{R}$ by
$||x|| = \sqrt{(x, x)}$.
Use the Cauchy-Bunyakowski-Schwarz (CBS) inequality to show that $||\cdot||$ is a norm on $X$.

Let f: X → Y be a function between metric spaces (X, d) and (Y, ρ), and let K ⊆ X.
(4.1) Let x0 ∈ X. What does it mean to say that f is continuous at the point x0.
(4.2) Prove that the following statements are equivalent.
(a) f is continuous.
(b) For every subset A ⊆ X, f(A) ⊆f(A).
(c) If B is closed in Y, then f^-1(B) is closed in X.
(d) If D is open in Y, then f^-1(D) is open in X.
(4.3) Define what K is a compact subset of X means and explain any terminology that you use in your definition.
(4.4) If f is continuous and K is compact, use the definition of compactness to show that f(K) is a compact subset of Y.
(4.5) Suppose that X is a vector space over the field of complex numbers ℂ with inner product (·, ·).
Define ||·||: X →ℝ by
||x|| = √((x, x)).
Use the Cauchy-Bunyakowski-Schwarz (CBS) inequality to show that ||·|| is a norm on X.

Added by Thomas O.

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