Let $X$ be a Banach space, and $B_1\supseteq B_2 \supseteq\cdots $ . Show that $\bigcap\limits_{i=1}^\infty B_i\neq\emptyset$
73462
Nested sequences of balls in a Banach space
113790
$\overline{B_1(0)}$ is not weakly* sequentially compact in $(l^\infty)$'
71682
Proving that $X$ is a Banach space iff convergence of $\sum\|x_n\|$ implies convergence of $\sum x_n$
158189
If $\{E_\alpha\}$ is connected, $\bigcap\limits_{\alpha\in A}E \neq \emptyset$, then $\bigcup\limits_{\alpha\in A}E$ is connected
59162
Prove $\liminf\limits_{n\to\infty} F(x_n)\le F(x)$
53883
If every absolutely convergent series is convergent then $X$ is Banach
139367
Hahn Banach Theorem: Transfinite Induction
96848
Prove: $\limsup_{n\to\infty} (a_n)\leq \sup(a_n)_{n=1}^\infty$
109632
Show reflexive normed vector space is a Banach space
62428
How to prove this inequality in Banach space?
97391
Prove that $(C^1[0,1], \|\cdot\|)$ is not a Banach space
91901
Prove that $\lim_{n\to\infty}a_n\le \lim_{n\to\infty}b_n$
126821
Let $f:[0,1]\to \mathbb{R}$. Show that $\{a \in [0,1]: f(a)\neq 0\} = \bigcup_{n=1}^{\infty} \{a \in [0,1]: |f(a)|\geq 1/n\}$
150896
Show that $\lim_{p \to \infty} \| f \|_p = \| f \|_\infty$
48593
Proof that if $s_n \leq t_n$ for $n \geq N$, then $\liminf_{n \rightarrow \infty} s_n \leq \liminf_{n \rightarrow \infty} t_n$
156727
A step in the proof of Characterization of $W^{1,\infty}$
104607
Hahn-Banach theorem, dual space