Intereting Posts

Probability that the convex hull of random points contains sphere's center
Graph (or Group) in Astronomy
A triangle with vertices on the sides of a square, with one at a midpoint, cannot be equilateral
On the commutator subgroup of a group
How can I give a bound on the $L^2$ norm of this function?
About the intersection of two countable sets
Justification of algebraic manipulation of infinitesimals
Using variation of parameters, how can we assume that nether $y_1$, $y_2$ equal zero?
When does the isomorphism $G\simeq ker(\phi)\times im(\phi)$? hold?
Proof of a trigonometric expression
Direct proof. Square root function uniformly continuous on $[0, \infty)$ (S.A. pp 119 4.4.8)
Find $\int_{0}^{\infty} \frac{\ln(x)}{1+x^2}dx$
Existence of Hamel basis, choice and regularity
Recursive square root problem
Euler numbers grow $2\left(\frac{2}{ \pi }\right)^{2 n+1}$-times slower than the factorial?

This question is taken from Saxe K -Beginning Functional Analysis.

Show that the closed unit ball in $C[0,1]$ is not compact by proving that it is not sequentially compact.

(It’s assumed that we are using the uniform norm).

- Topological spaces in which every proper closed subset is compact
- Is the Alexandroff double circle compact and Hausdorff?
- Given a fiber bundle $F\to E\overset{\pi}{\to} B$ such that $F,B$ are compact, is $E$ necessarily compact?
- Using functions to separate a compact set from a closed set in a completely regular space
- Is every compact space compactly generated?
- Prob. 5, Sec. 27 in Munkres' TOPOLOGY, 2nd ed: Every compact Hausdorff space is a Baire space

I’ve been working on this for ages but I could not come up with any sequence $\{f_n\}$ in the unit ball such that there exists $N\in \mathbb{N}$ such that for all $m,n\geq N$ we have that $d(f_n,f_m)>c$. Should be a nice example of this, please help me!

- How to prove convergence of polynomials in $e$ (Euler's number)
- Convergence in measure implies convergence almost everywhere of a subsequence
- Convergence of $\sum_n \frac{n!}{n^n}$
- Selection of $b_n$ in Limit Comparison Test for checking convergence of a series
- A sequence of functions converging to the Dirac delta
- perfect map in topology
- Convergence of a product series with one divergent factor
- Limit of sequence of growing matrices
- Limit point of an infinite set in a compact space
- proof Intermediate Value Theorem

Consider $f_n(t)=t^n$, $0\le t\le 1$. Then $\{f_n\} \subset \overline{B(0,1)}$ (closed unit ball), but no subsequence of $\{f_n\}$ converges in $C[0,1]$ (with the sup norm).

**Hint:** each subsequence should converge uniformly to the pointwise limit, which is not continuous.

So take any bounded sequence in $C[0,1]$ which converges pointwise to a non-continuous function.

- Tensor product of monoids and arbitrary algebraic structures
- a question related to two competing patterns in coin tossing
- Quotient rings of Gaussian integers
- Sign of Laplacian at critical points of $\mathbb R^n$
- Evaluation of $\lim_{n\rightarrow \infty}\sum_{k=1}^n\sin \left(\frac{n}{n^2+k^2}\right)$
- If $f:\mathbb{R}\to\mathbb{R}$ is a left continuous function can the set of discontinuous points of $f$ have positive Lebesgue measure?
- Counting two ways, $\sum \binom{n}{k} \binom{m}{n-k} = \binom{n+m}{n}$
- Show $(W_1 + W_2 )^\perp = W_1^\perp \cap W_2^\perp$ and $W_1^\perp + W_2^\perp ⊆ (W_1 \cap W_2 )^\perp$
- Evaluate $\int_0^{\pi} \frac{\sin^2 \theta}{(1-2a\cos\theta+a^2)(1-2b\cos\theta+b^2)}\mathrm{d\theta}, \space 0<a<b<1$
- What is the “Principle of permanence”?
- Help with $\lim_{x, y\to(0, 0)} \frac{x^2y}{x^4+y^2}?$
- Isomorphic Free Groups and the Axiom of Choice
- Matrix Identity
- Eigenvalue problems for matrices over finite fields
- Multiple self-convolution of rectangular function – integral evaluation