# Finite codimension

Let $E,F$ be Banach (more generally FrÃ©chet) spaces and let $T,S:E\to F$ be continuous linear maps. Suppose that $T$ is surjective and $S$ is compact. Prove that $T+S$ has closed image and $\dim \mathrm{im}(T+S)<\infty$.

# Too large to measure

Let $X$ be an infinite dimensional normed space. There is no non-trivial translation invariant Borel measure on $X$ which is finite on open balls.

# Spooky geometry IV

Let $[a,b]$ and $[c,d]$ be subintervals of $[0,1]$ and call themÂ non-overlapping if they intersect only possibly in their endpoints. Construct a continuous curve $\sigma:[0,1]\to H$ with $H$ a Hilbert space, such that if $[a,b]$ and $[c,d]$ are any non-overlapping subintervals of $[0,1]$, then $\sigma(b)-\sigma(a)$ and $\sigma(d)-\sigma(c)$ are orthogonal.

Harder: show that this can be carried out in any infinite-dimensional Hilbert space.

# Universal and injective Banach spaces

Prove the following remarkable properties of $\ell_1$ and $\ell_\infty$

1. Given any normed space $X$ and a subspace $Y$, any continuous linear map $T:Y\to\ell_\infty$ may be extended to all of $X$ preserving its norm.
2. Every separable Banach space is a quotient of $\ell_1$