Question no $1$ > Let $D$ denote the closed unit disc and let $S^1$ denote the unit circle in $\mathbb{R}^2$Let $E=\{(x,y)\in \mathbb{R}^2|\frac{x^2}{4}+\frac{y^2}{9}\le 1\}$ Which of the following statements are true?
a)If $f:E\to E$ is continuous, than there exists $x\in E$ such that $f(x)=x$.
b)If $f:D\to S^1$ is continuous, than there exists $x\in S^1$ such that $f(x)=x$.
c)If $f:S^1\to S^1$ is continuous, than there exists $x\in S^1$ such that $f(x)=x$.
Question no - 2 > Let V and W be normed linear spaces and let T : V → W be a continuous linear operator. Let B be the closed unit ball in V . In which of the following cases is T(B) compact?
$a.$ $V = C_1[0,1]$,$W = C[0,1]$ and $T(f) = f$.
b. $V = W = `_2$ and $T(x) = (0,x_1,x_2,···)$, where $x = (x_n) ∈ `2$.
c. $V = W = `2$ and $T(x) = (x_1,x_2,···,x_10,0,···,0,···)$, where $x = (x_n) ∈ `2$.
Question no 3> For $x = (x_n) ∈ `2$, define $T(x) = (0,x_1,x_2,···)$ and $S(x) = (x_2,x_3,···)$. Which of the following statements are true?
a. norm($T$) = norm($S$) = $1$.
b. If $A : `2 → `2$ is a continuous linear operator such that norm$(A−T)$ < $1$, then $SA$ is invertible.
c. If $A$ is as above, then $A$ is not invertible.
Actually these questions were given in Topology Section of a PH.D screening Test Question. I have read up to Local Compactness from Mukresh. But I don't think I am eligible to solve those questions. Can you please let me know how much should I read from Mukresh to be able to answer those questions?
It will help me to set my target. Thank You in Advance.