Linear algebra (Osnabrück 2024-2025)/Part II/Exercise sheet 52

Exercises

Exercise

Let ${}(M,d)$ be a metric space. Show that the open balls ${}U{\left(x,\epsilon \right)}$ are open.

Exercise

Let ${}(M,d)$ be a metric space. Show that the closed balls ${}B\left(x,\epsilon \right)$ are closed.

Exercise

Let ${}(M,d)$ be a metric space, and let ${}P\in M$ be a point. Show that ${}\{P\}$ is closed.

Exercise

Let ${}(M,d)$ be a metric space. Show that the following properties hold.

1. The empty set ${}\emptyset$ and the total space ${}M$ are open.
2. Let ${}I$ be an arbitrary index set, and let ${}U_{i}$, ${}i\in I$, denote open sets. Then also the union

   $\bigcup _{i\in I}U_{i}$

   is open.

3. Let ${}I$ be a finite index set, and let ${}U_{i}$, ${}i\in I$, be open sets. Then also the intersection

   $\bigcap _{i\in I}U_{i}$

   is open.

Exercise

Let ${}X$ be a Hausdorff space, and let ${}Y\subseteq X$ be a subset that carries the induced topology. Let ${}Y$ be compact. Show that ${}Y$ is closed in ${}X$.

Exercise

Let ${}(M,d)$ be a metric space, and ${}m\in M$. Show that the constant mapping

$f\colon L\longrightarrow M,x\longmapsto m,$

is continuous.

Exercise

Let ${}(M,d)$ be a metric space. Show that the identity

$M\longrightarrow M,x\longmapsto x,$

is continuous.

Exercise

Let ${}(M,d)$ be a metric space, and let ${}T\subseteq M$ denote a subset, equipped with the induced metric. Show that the inclusion ${}T\subseteq M$ is continuous.

Exercise

Let ${}V$ be a normed ${}{\mathbb {K} }$-vector space, and let

$\varphi _{w}\colon V\longrightarrow V,v\longmapsto v+w,$

denote the translation with the vector ${}w\in V$. Show that ${}\varphi _{w}$ is continuous.

Exercise

Let ${}(M,d)$ be a metric space, and let

$f\colon M\longrightarrow \mathbb {R}$

denote a continuous function. Let ${}x\in M$ be a point with ${}f(x)>0$. Show that also ${}f(y)>0$ holds for all ${}y$ from an open ball neighbourhood of ${}x$.

Exercise

Let ${}(M,d)$ be a metric space, and let ${}a<b<c$ be real numbers. Let

$f\colon [a,b]\longrightarrow M$

and

$g\colon [b,c]\longrightarrow M$

be continuous mappings with ${}f(b)=g(b)$. Show that the mapping

$h\colon [a,c]\longrightarrow M$

given by

$h(t)=f(t){\text{ for }}t\leq b{\text{ and }}h(t)=g(t){\text{ for }}t>b$

is also continuous.

Exercise

Show that the addition

${\mathbb {K} }\times {\mathbb {K} }\longrightarrow {\mathbb {K} },(x,y)\longmapsto x+y,$

and the multiplication

${\mathbb {K} }\times {\mathbb {K} }\longrightarrow {\mathbb {K} },(x,y)\longmapsto x\cdot y,$

are continuous.

Exercise

Show that a polynomial function

$f\colon \mathbb {R} ^{n}\longrightarrow \mathbb {R} ,(x_{1},\ldots ,x_{n})\longmapsto f(x_{1},\ldots ,x_{n}),$

is continuous.

Exercise

Show that a real quadric, that is, the zero set given by a real polynomial of degree two, is a closed subset of ${}\mathbb {R} ^{n}$.

Does that also hold for the zero set of a polynomial of higher degree?

Exercise

Let ${}L,M,N$ be metric spaces, and let

$f:L\longrightarrow M\,\,{\text{ and }}\,\,g:M\longrightarrow N$

mappings. Suppose that ${}f$ is continuous in ${}x\in L$, and that ${}g$ is continuous in ${}f(x)\in M$. Show that the composition

$g\circ f\colon L\longrightarrow N,x\longmapsto g(f(x)),$

is continuous in ${}x$.

Exercise

Show that the function

$\mathbb {C} \longrightarrow \mathbb {C} ,z\longmapsto \vert {z}\vert ,$

is continuous.

Exercise

Show that the function

$f\colon \mathbb {R} ^{2}\longrightarrow \mathbb {R} ,(x,y)\longmapsto {\max {\left(x,y\right)}},$

is continuous.

Exercise

We consider the function

$f\colon \mathbb {R} ^{2}\longrightarrow \mathbb {R}$

given by

${}f(x,y):={\begin{cases}0\,,{\text{ if }}x\leq 0\,,\\0\,,{\text{ if }}y\leq 0\,,\\y/x\,,{\text{ if }}x\geq y>0\,,\\x/y\,,{\text{ if }}y>x>0\,.\end{cases}}\,$

Show that the restriction of ${}f$ to every line that is parallel to the ${}x$-axis or to the ${}y$-axis is continuous, but ${}f$ itself is not continuous.

Exercise

Let ${}(M,d)$ be a metric space, and let ${}T\subseteq M$ denote a notleere subset. Show that via

${}d_{T}(x):=\inf {\left(d(x,y),\,y\in T\right)}\,$

we get a well-defined continuous function ${}M\rightarrow \mathbb {R}$.

Exercise

Let ${}V$ be an infinite-dimensional normed ${}\mathbb {R}$-vector space. Show that there exists a linear mapping

$V\longrightarrow \mathbb {R}$

that is not continuous.

Exercise

Let ${}(M,d)$ be a metric space, and let ${}{\left(x_{n}\right)}_{n\in \mathbb {N} }$ denote a sequence in ${}M$. Show that the sequence converges in the sense of a metric space if and only if the sequence converges in the sense of topology.

Exercise

Let ${}X$ and ${}Y$ be topological spaces, and let

$\varphi \colon X\longrightarrow Y$

denote a continuous mapping. Let ${}X$ be compact. Show that the image ${}\varphi (X)\subseteq Y$ is also compact.

Exercise

Show that the open unit interval ${}]0,1[$ and the closed unit interval ${}[0,1]$ are not homeomorphic.

Exercise

Show that the mapping

$[0,2\pi [\longrightarrow S^{1},t\longmapsto \left(\cos t,\,\sin t\right),$

between the half-open interval ${}[0,2\pi [$ and the unit circle

${}S^{1}={\left\{P\in \mathbb {R} ^{2}\mid \Vert {P}\Vert =1\right\}}\,$

is continuous and bijective, and that the inverse mapping is not continuous.

For an arbitrary set ${}M$, we can define a metric via

${}d(x,y):={\begin{cases}0,\,&{\text{ if }}x=y\,,\\1,\,&{\text{ if }}x\neq y\,.\end{cases}}\,\,$

This is called the discrete metric.

Exercise

Let ${}X=\mathbb {R} ^{n}$, equipped with the Euclidean metric, and ${}Y=\mathbb {R} ^{n}$, equipped with the discrete metric. Let

$f\colon Y\longrightarrow X$

be the identity. Show that ${}f$ is continuous but the inverse mapping ${}f^{-1}$ is not continuous.

Exercise

Let ${}X$ be a nonempty set equipped with the discrete metric. Show that a continuous mapping

$f\colon \mathbb {R} \longrightarrow X$

is constant.

Exercise

Let ${}{\mathbb {K} }=\mathbb {R}$ or ${}=\mathbb {C}$. Let ${}H\subset {\mathbb {K} }^{n+1}$ be an ${}n$-dimensional affine subspace that does not contain the origin, and let ${}{\tilde {H}}$ denote the linear subspace parallel to ${}H$. Let ${}U\subseteq H$ be a subset that is open in ${}H\cong {\mathbb {K} }^{n}$

(in the metric topology), and let ${}V$ denote the union of all lines through the origin and through a point of ${}U$. Show that the intersection of ${}V$ with ${}{\mathbb {K} }^{n+1}\setminus {\tilde {H}}$ is open.

Hand-in-exercises

Exercise (2 marks)

Let ${}V\subseteq \mathbb {R} ^{n}$ be a linear subspace in the Euclidean space ${}\mathbb {R} ^{n}$. Show that ${}V$ is closed in ${}\mathbb {R} ^{n}$.

Exercise (4 marks)

Let ${}V$ be a Euclidean space. Show that the norm

$V\longrightarrow \mathbb {R} ,v\longmapsto \Vert {v}\Vert ,$

is a continuous mapping.

Exercise (4 marks)

Let

$\varphi \colon \mathbb {R} ^{n}\longrightarrow \mathbb {R} ^{m}$

be continuous and additive, that is, we have ${}\varphi (x+y)=\varphi (x)+\varphi (y)$ for all ${}x,y\in \mathbb {R} ^{n}$. Show that ${}\varphi$ is ${}\mathbb {R}$-linear.

Exercise (5 marks)

Suppose that in the origin ${}0\in \mathbb {R} ^{3}$, there is the pupil of an eye (or just a small hole), and in the plane determined by ${}x=-1$, we have a retina ${}N\cong \mathbb {R} ^{2}$ (or a photographic plate). Determine the mapping

$\mathbb {R} _{+}\times \mathbb {R} \times \mathbb {R} \longrightarrow \mathbb {R} ^{2}$

that describes the natural vision (or taking a photo) (that is, a point of the half space is mapped through the pupil to a point of the retina). Is this mapping continuous? It is linear?

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