Questions tagged [vector-spaces]
For questions about vector spaces and their properties. More general questions about linear algebra belong under the [linear-algebra] tag. A vector space is a space which consists of elements called "vectors", which can be added and multiplied by scalars
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Assume that for every $f\in X^*$, there exists $y \in X$ such that $f(x)=\langle x, y \rangle$ for every $x \in X$. Show that $X$ is a complete space.
Let $(X, \langle \cdot, \cdot \rangle)$ be a real or complex vector space with an inner product. Assume that for every $f \in X^*$, there exists $y \in X$ such that $f(x) = \langle x, y \rangle$ for ...
user1315539
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Complexification of a vector space $V$
The tensor product $V \otimes \mathbb{C}$ is formed by taking the real vector space $V$ (where $\dim V=n$) and extending its scalars from $ \mathbb{R} $ to $ \mathbb{C} $. Elements in $ V \otimes \...
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Intution behind $\mathrm{Hom}(U\otimes V, W) \cong \mathrm{Hom}(U, \mathrm{Hom}(V, W)). $
I'm currently self-studying Tensor products and came across this result:
$$
\mathrm{Hom}(U\otimes V, W) \cong \mathrm{Hom}(U, \mathrm{Hom}(V, W)).
$$
Whilst I can follow the proof algebraically, I can'...
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Question about Artin's Algebra Example 3.3.4 on vector space
Example 3.3.4 Let $F$ be the prime field $\mathbb{F_p}$. The space $F^2$ contains $p^2$ vectors, $p^2-1$ of which are nonzero. Because there are $p-1$ nonzero scalars, the subspace $W=\{cw\}$ spanned ...
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Inequality relating quotient norm and norm
In an comment under an answer to this question
How to show that quotient space $X/Y$ is complete when $X$ is Banach space, and $Y$ is a closed subspace of $X$?
it is claimed that we have the ...
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Criterion for products of projections.
Given an arbitrary vector space $V$ and two (linear) projections $P$ and $Q$.
In the case of orthogonal projections, it is relatively easy to show that the product $PQ$ is a projection if and only if $...
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$\|f^{-1} \|= \frac{1}{a} $
We define
$$B(X,Y)= \{ f \mid f:X \to Y, \text{ is continuous and linear function}\}$$
Let $X,Y$ bee Banach normed spaces, $f \in B(X,Y)$, $a>0$, and $ \|f(x)\| \ge a\|x\|$ for all $x\in X$.
Then
$...
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For normed spaces $ E=\{x\in X \mid \inf\{ \|x-e\| \mid e \in E \}=0 \} $
Let X be a normed vector space and $ E \subset X$
Prove that $$ E=\{x\in X \mid \inf\{ \|x-e\| \mid e \in E \}=0 \} $$
I tried to prove like this:
Let
$\begin{align*}
x \in E &\Rightarrow 0 \le \...
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Taylor Expansion of a vector-valued function with 2 vectors as input
Let a function $f(x,u): \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}^n$.
I wonder how to expand it around $(x_n, u_n)$.
For the time being, keeping it only up to the first order is enough
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About strictly convex norm
Let X be a normed vector space. $\| . \|$ is a norm.
we said this norm is a strictly convex norm if $$ \forall x,y \in X : \| x\| \le 1, \|y\| \le 1 \Rightarrow \| \frac{x+y}{2} \| <1 $$
I have ...
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$\|p\| = \max_{n \in \mathbb{N}} |a_n p^{(n)}(0)|$, a norm is defined on $X$, where $p^{(n)}$ is the nth derivative of the polynomial $p$
Let $X$ be the vector space of all polynomials with real coefficients and let $(a_0, a_1, \ldots)$ be a sequence of positive real numbers. I was able to show that with the prescription $\|p\| = \max_{...
user1316790
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Let $X$ be the vector space of all real sequences that have at most finitely many non-zero terms. Is $(X, \| \cdot \|)$ a Banach space?
Let $X$ be the vector space of all real sequences that have at most finitely many non-zero terms. I was able to show that the prescription $\| \{x_n\}_{n \in \mathbb{N}} \| = \max_{n \in \mathbb{N}} |...
user1316777
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Find the plane passing through $A$ and tangent with a sphere $(S)$ such that the distance between $B$ and this plane is greatest.
In the $Oxyz$ space, consider point $A(0;8;2)$ and the sphere $(S): (x-5)^2 + (y+3)^2 + (z-7)^2 = 72$ and point $B(9;-7;23)$. $(P)$ is the plane passing through $A$ and tangent with $(S)$ such that ...
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Cartesian and parametric representation of a vector subspace, given a base
I have this definition from my teacher's notes:
"Let $\phi_{\mathcal{B}}:V \rightarrow \mathbb{K}^n$ an isomorphism such as $\phi_{\mathcal{B}} (\textbf{v})=(\textbf{v})_{\mathcal{B}}$, i.e. $\...
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Why must a reducible KG-module of dimension 2 have a KG-submodule of dimension 1?
I'm trying to see this with the Quaternion group $Q_8 = \langle a,b \ | \ a^2=b^2,\ bab^{-1} = a^{-1} \rangle $, and the representation $\rho: Q_8 \rightarrow GL_2 (\mathbb{C}) $ defined by $\rho(a)=...
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