$B=\{f \in C[0,1] : \sup_{x\in [0,1]} |f(x)| \leq 1\}$, $\Gamma \in C^{*}$ but $\Gamma(B)$ is open
65664
Separation theorem on the space of all complex continuous functions
133870
if $\alpha\leq\beta$ then there is $\gamma$ such as $\alpha+\gamma=\beta$
17343
Function on $[a,b]$ that satisfies a Hölder condition of order $\alpha > 1 $ is constant
68409
Consider $f(x,y)=\sup_{θ,φ}\{||e^{i θ }x+ e^{i φ }y||^2: θ,φ∈R\}$ in which $x,y\in\mathbb C^n$
29849
How can I show that $\sup(AB)\geq\sup A\sup B$ for $A,B\subset\mathbb{R}$ where $A\cup B$ is positive and bounded?
136789
How to prove $a \lt b$ for all $a \in A, b \in B$ implies $\sup A \leq \inf B$?
70264
Prove that the inverse image of an open set is open
79424
Don't understand proof that interior of a set is open
152817
Proving $\sup B=\inf A$
143354
Prove that there exists $c\in\mathbb{R}$ such that $f'(c)=f(c)$
92774
Prove that a linear subspace of $C([0,1])$ is closed
14516
The space of continuous functions $C([0,1])$ is not complete in the $L^2$ norm
16654
Unions and Intersections of Open Sets are Open
126821
Let $f:[0,1]\to \mathbb{R}$. Show that $\{a \in [0,1]: f(a)\neq 0\} = \bigcup_{n=1}^{\infty} \{a \in [0,1]: |f(a)|\geq 1/n\}$
135577
Prove the existence of $c$ such that $f'(c) = 2c(f(c) - f(0))$
130537
Proving that $d(f,g)=\|f-g\| = \sup \limits_{0\leq x \leq 1} |f(x)-g(x)|$ is a metric on $X=C[0,1]$
118903
The sup norm on $C[0,1]$ is not equivalent to another one, induced by some inner product