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Suppose that $X$ is a nonempty complete metric space and $T:X\to X$ is a contraction. Then $T$ has a unique fixed point.

Suppose that $V$ and $W$ are normed vector spaces and $T:V\to W$ is linear. Then $T$ is continuous if and only if $T$ is bounded.

Suppose that $V$ is a normed vector space and $W$ is a Banach space. Then the space $ℬ(V,W)$ of bounded linear maps from $V$ to $W$, with the operator norm, is a Banach space.

A complete metric space cannot be written as a countable union of closed sets with empty interior.

Suppose that $V$ and $W$ are Banach spaces and $T:V\to W$ is a surjective bounded linear map. Then $T$ maps open subsets of $V$ to open subsets of $W$.

Suppose that $V$ and $W$ are Banach spaces and $T:V\to W$ is linear. If the graph of $T$ is closed in $V\times W$, then $T$ is bounded.

Suppose that $V$ is a Banach space, $W$ is a normed vector space, and $𝒜\subseteq ℬ(V,W)$. If ${\sup }_{T\in 𝒜}\Vert Tv\Vert <\infty$ for every $v\in V$, then ${\sup }_{T\in 𝒜}\Vert T\Vert <\infty$.