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Moderate

# Cauchy Sequences and Functionals

TOPO-FNWOYP

Let $(X,\mathcal{T})$ be a topological vector space. Assume that $\mathbb{R}$ is equipped with the usual Euclidean topology. Let $f:X\to \mathbb{R}$ be a map such that for any $a,b \in X$, we have $|f(a) - f(b)| \leq | f(a-b)|.$ Let $e$ be the zero vector in $X$. Assume that $f(e)=0$, and that $f$ is continuous at $e$. Let $(x_n)_n$ be a Cauchy sequence in $(X,\mathcal{T})$.

Which of the following MUST be true? Select ALL that apply.

A

The vector addition is a continuous map from $X\times X\,$ to $X$.

B

The scalar multiplication may not be a continuous map from $\mathbb{R} \times X$ to $X$.

C

$(f(x_n))_n$ is a Cauchy sequence in $\mathbb{R}$.

D

$(f(x_n))_n$ is a convergent sequence in $\mathbb{R}$.

E

$f$ is a linear map, or equivalently, $f(\alpha x+\beta y) = \alpha f(x) + \beta f(y)$, for all $\alpha,\beta\in \mathbb{R}$ and all $x,y\in X$.

F

$f$ is a continuous map from $X$ to $\mathbb{R}$.