Quotient space and continuous operator Let $X,Y$ be normed spaces and $T \in L(X,Y)$. We define
$K=\ker(T)=\{x \in X:Tx=0\}$.
Show that for $U:X/K \rightarrow Y, x+K \rightarrow Tx$:
$U\in L(X/K,Y)$.
Well, by definition a linear operator is continous if it is continous at 0 (in the domain). So I'd need to show that $U$ is continuous at $0 \in X/K$ but this is where I'm stuck. I have read other solutions using the norm but I don't really understand how one could define the norm on a quotient space.
 A: Actually, if you want to show that $U$ is continuous you need to equip the space $X/K$ with some topology and the canonic topology on $X/K$ is that which is induced by the so called quotient-norm:
Consider the following map
$$
\Vert \cdot \Vert_{X/K} \colon X/K \to [0, \infty), \quad \lVert x + K \rVert_{X/K} := \inf \{\lVert x - y \rVert : y \in K\}.
$$
It follows from the definition that this map is well-defined (why?).
Let us show that this map is a norm. Suppose $x + K \in X/K$ is such that $\lVert x+ K \rVert_{X/K} = 0$. Then there exists a sequence $(y_n)_n$ in $K$ such that $\lVert x - y_n \rVert \to 0$ for $n \to \infty$. Since $K$ is closed, it follows that $x \in K$ and therefore $x + K = 0 + K$.
Now let $\lambda \in \mathbb C$. Then we have
\begin{align*}
\lVert \lambda(x + K) \rVert_{X/K} &= \lVert \lambda x + K \rVert_{X/K} = \inf \{\lVert \lambda x - y \rVert : y \in K\} \\ &= \lvert \lambda \rvert \inf \{\lVert x - y \rVert : y \in K\} = \lvert \lambda \rvert \cdot \lVert x + K \rVert_{X/K}.
\end{align*}
Finally, if $y + K \in X/K$, then
\begin{align*}
\lVert (x + K) + (y + K) \rVert_{X/K} &= \lVert x + y + K \rVert_{X/K} = \inf \{\lVert x + y - z \rVert : z \in K\} \\ &\leq \inf \{\lVert x - z \rVert + \lVert y - z \rVert : z \in K\} \\
&= \inf \{\lVert x - z \rVert : z \in K\} + \inf \{\lVert y - z \rVert : z \in K\} \\
&= \lVert (x + K) \rVert_{X/K} + \lVert (y + K) \rVert_{X/K}.
\end{align*}
Hence, $\Vert \cdot \Vert_{X/K}$ is a norm on $X/K$. Can you take it from here?
