Recent content by complexnumber

Is this correct? [PLAIN]http://9ya7ng.blu.livefilestore.com/y1pFP94kdkanTL7oWHQXuecgsG7MYfNfM3fHYVt7AE01cgDtbQY8VkjQk94V8H5WceDMp8kOlh-X1WSs79GZtIUTEWTFMCBtceU/q3.jpg

Homework Statement For the Banach space X = C[0,1] with the supremum norm, fix an element g \in X and define a map \varphi_g : X \to \mathbb{C} by \begin{align*} \varphi_g(h) := \int^1_0 g(t) h(t) dt, \qquad h \in X \end{align*} Define W := \{ \varphi_g | g \in X \}. Prove...

I think the question must be assuming we are using the extended real line. In my lecture notes the \infty measure is allowed. Is my answers correct assuming extended real line?

Homework Statement Given a measure space (\mathbb{R}, \mathcal{B}(\mathbb{R}),\mu) define a function F_\mu : \mathbb{R} \to \mathbb{R} by F_\mu(x) := \mu( (-\infty,x] ). Prove that F_\mu is non-decreasing, right-continuous and satisfies \displaystyle \lim_{x \to -\infty} F_\mu(x) = 0. (Right...

Homework Statement Given a \sigma-algebra (X,\mathcal{A}), let f_n : X \to [-\infty,\infty] be a sequence of measurable functions. Prove that the set \{ x \in X | \lim f_n (x) \text{ exists} \} is in \mathcal{A}. Homework Equations Let (X,\mathcal{A}) be a \sigma-algebra and...

U \in tau can be any open interval in \mathbb{R} and contain infinite number of rational numbers, but when it intersects with a set of rational numbers, the resulting set contains infinite number of rational points. You were right. I didn't think what U is.

Since S only contain closed single rational points, and the definition of subspace topology is \tau_S = \{ S \cap U | U \in \tau \}, shouldn't the subspace topology only contain single points and not (q-\epsilon, q + \epsilon)?

S_1 is not closed because the function f = 0 is a limit point outside S_1. Therefore S_1 is not compact. For S_2, the metric space d_\infty(f,g) := \norm{f - g}_\infty means that it is bounded, however it does not make S_2 equicontinuous. Is the subset closed?

Homework Statement Identify the compact subsets of \mathbb{Q} \cap [0,1] with the relative topology from \mathbb{R}. Homework Equations The Attempt at a Solution Is it all finite subsets of \mathbb{Q} \cap [0,1]? The relative topology contains single rational points in [0,1]...

Homework Statement Let (X,d) = (C[0,1], d_\infty), S_1 is the set of constant functions in B(0,1), and S_2 = \{ f \in C[0,1] | \norm{f}_\infty = 1\}. Are S_1 and S_2 compact? Homework Equations The Attempt at a Solution I am trying to use the Arzela - Ascoli theorem. For S_1, the set of...

Thank you very much for detailed explanation about homeomorphic sets. I did not know those rules and thought the question is asking which set is homeomorphic to the complex space \mathbb{C}. None of these sets are homeomorphic to each other. Is it correct?

Homework Statement Consider the following subsets of \mathbb{C}, whose descriptions are given in polar coordinates. (Take r \geq 0 in this question.) \begin{align*} X_1 =& \{ (r,\theta) | r = 1 \} \\ X_2 =& \{ (r,\theta) | r < 1 \} \\ X_3 =& \{ (r,\theta) | 0 < \theta < \pi, r > 0 \}...

Homework Statement 1. Prove that if a metric space (X,d) is separable, then (X,d) is second countable.2. Prove that \ell^2 is separable. Homework Equations The Attempt at a Solution 1. \{ x_1,\ldots,x_k,\ldots \} is countable dense subset. Index the basis with rational numbers, \{ B(x,r) | x...

Why is \{A_n\}_{n\in\mathbb{N}}^{\infty} non-empty? Do you need to prove it or is it obvious? I am working on the same problem and cannot figure out this part.

Homework Statement Let (X,\tau) be a compact Hausdorff space, and let f : X \to X be continuous, but not surjective. Prove that there is a nonempty proper subset S \subset X such that f(S) = S. [Hint: Consider the subspaces S_n := f^{\circ n}(X) where f^{\circ n} := f \circ \cdots \circ...