# $d(w, V) = 1$ and $U = V \oplus\{\lambda w :\lambda \in \mathbb{F}\}$.

Let U be a normed space, $w \in U$ and $\|w\| = 1$. Show that $U$ has a closed
subspace $V$ such that $d(w, V) = 1$ and $U = V \oplus\{\lambda w :\lambda \in \mathbb{F}\}$.

#### Solutions Collecting From Web of "$d(w, V) = 1$ and $U = V \oplus\{\lambda w :\lambda \in \mathbb{F}\}$."

As we know by Hahn-Banach theorem $\exists$ a bounded linear functional $f$ on $U$ such that $f(w)=||w||$ and $||f||=1$. Consider $V=Ker(f)$. As $f$ is continuous $V$ will be closed. It is easy to see that $V\cap L\{w\}=\{0\}$. As $Ker(f)$ is of co-dimensional $1$ and $w\notin V$, then $U=V\bigoplus L\{w\}$.

Now it remains to show that $d(w,V)=1$.

1. $d(w,V)\leq ||w-0||=1$ (as $0$ is always in $Ker(f)$).

2. Now $1=||w||=f(w)=f(w-v)$, for any $v\in V$. As $f$ is bounded with $||f||=1$, so $1\leq ||w-v||$, for any $v\in V$. So $d(w,V)=inf_{v\in V}||w-v||\geq 1$.

Note: $L\{w\}=\{\lambda w: \lambda \in \mathbb{F}\}$.
The text passage contains a mathematical problem statement and a detailed solution to that problem. The problem is to show the existence of a closed subspace \( V \) in a normed space \( U \) with specific properties, and the solution uses the Hahn-Banach theorem to construct such a subspace.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# $d(w, V) = 1$ and $U = V \oplus\{\lambda w :\lambda \in \mathbb{F}\}$.

Let U be a normed space, $w \in U$ and $\|w\| = 1$. Show that $U$ has a closed
subspace $V$ such that $d(w, V) = 1$ and $U = V \oplus\{\lambda w :\lambda \in \mathbb{F}\}$.

#### Solutions Collecting From Web of "$d(w, V) = 1$ and $U = V \oplus\{\lambda w :\lambda \in \mathbb{F}\}$."

As we know by Hahn-Banach theorem $\exists$ a bounded linear functional $f$ on $U$ such that $f(w)=||w||$ and $||f||=1$. Consider $V=Ker(f)$. As $f$ is continuous $V$ will be closed. It is easy to see that $V\cap L\{w\}=\{0\}$. As $Ker(f)$ is of co-dimensional $1$ and $w\notin V$, then $U=V\bigoplus L\{w\}$.

Now it remains to show that $d(w,V)=1$.

1. $d(w,V)\leq ||w-0||=1$ (as $0$ is always in $Ker(f)$).

2. Now $1=||w||=f(w)=f(w-v)$, for any $v\in V$. As $f$ is bounded with $||f||=1$, so $1\leq ||w-v||$, for any $v\in V$. So $d(w,V)=inf_{v\in V}||w-v||\geq 1$.

Note: $L\{w\}=\{\lambda w: \lambda \in \mathbb{F}\}$.
user
The text passage contains a mathematical problem statement and a detailed solution to that problem. The problem is to show the existence of a closed subspace \( V \) in a normed space \( U \) with specific properties, and the solution uses the Hahn-Banach theorem to construct such a subspace.

TYPE: SINGLE_QUESTION_WITH_ANSWER