Questions tagged [direct-sum]
For questions about taking the direct sum of groups and other algebraic structures.
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First syzygy is unique up to direct summand
Let $R:=K[x_0,...,x_n]$ be the polynomial ring over a field $K$ and $M$ be finitely generated $R$ module. Let $(m_i)_{i=1,...,k}$ be a generating set for $M$. Then we can define the free module $F_0 :=...
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Sum of subspaces $V$ and $W$ of $\mathbb{R}^n$ with $\text{dim}(V)+\text{dim}(W)=n$
If $V$ and $W$ are subspaces of $\mathbb{R}^n$ such that $\text{dim}(V)+\text{dim}(W)=n$ and $V\cap W=\{\mathbf{0}\}$, can we immediately say that \begin{equation} V\oplus W=\mathbb{R}^n ?\end{...
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functors preserve isomorphism of direct sum.
In this proof in the Stacks project, it is mentioned that the decomposition of identity morphism of direct sum:
... because the composition $F(A) \oplus F(B) \xrightarrow{\varphi} F(A \oplus B) \...
- 301
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Isometric shift in disc algebra
Let $A$ be the disk algbra: Collection of all continuous functions on the closed unit disk which are analytic on the open unit disk, equipped with the supremum norm.
Now, consider the multiplication ...
- 1,350
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The functor that preserves the direct sum is an additive functor
I am learning about additive categories on the Stacks Project, and I encountered some difficulties in proving that functor that preserves direct sums is an additive functor.
There is such a ...
- 301
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A direct sum decomposition [duplicate]
I have tried to solve this problem. I am studying for an exam.
Let $V$ be a real finite-dimensional vector space and let $T \colon V \rightarrow V$ be and operator such that for some positive integer $...
- 145
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Prove that $\mathbb{F}^n=U+W$
Let $\mathbb{F}$ be a field.
$
\begin{matrix}U:=\left\{ \begin{pmatrix}a_{1}\\
a_{2}\\
\vdots\\
a_{n}
\end{pmatrix}\in\mathbb{F}^{n}\mid a_{1}=a_{2}=\cdots=a_{n}\right\} \\
W:=\left\{ \begin{pmatrix}...
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Let $ H $ be a Hilbert space and let $ A \in \mathcal{B}(H) $. Show that $ H = \ker A \oplus \overline{\operatorname{Im} A^*}. $
Let $ H $ be a Hilbert space and let $ A \in \mathcal{B}(H) $. Show that
$ H = \ker A \oplus \overline{\operatorname{Im} A^*}. $
Attempt: Since $ \operatorname{Im} A^* \subseteq \overline{\...
user1315539
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Proof of Krull schmidt theorem
I have been trying to understand the proof of this theorem from the book "Algèbres et modules." Cours et exercices.
I've been trying for a while, but I haven't been able to understand the ...
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Confusion about representing elements as pure tensors in the definition of the tensor product of dg-algebras
I'm refamiliarizing myself with the tensor product of dg-algebras and struggling to reconcile some of the basic definitions. Let $(A, d_A)$ and $(B, d_B)$ be dg-algebras over a field $k$, and assume ...
- 1,190
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What is $\bigoplus_{n\in \mathbb{N}} I^{n}$, is topology is Tychonoff subspace topology?
I want to know what is $\bigoplus_{n\in \mathbb{N}} I^{n}$ with $I=[0,1]$.
Im not sure if is just:
$$\bigoplus_{n\in \mathbb{N}} I^{n}=\{(x_{n})\in \prod_{n\in \mathbb{N}}I^{n}\mid \exists ~M\...
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Find subspace $T$ of $\mathbb{R}^{3}$ such that $\mathbb{R}^{3} =V \oplus T$
\begin{array}{l} V=\{( a+2b,2a+8b+2c,a+10b+4c) \ |\ a,b,c\in \mathbb{R}\}\\ =\{a( 1,2,1) +b( 2,8,10) +c( 0,2,4) \ |\ a,b,c\in \mathbb{R}\}\\ =Sp\{( 1,2,1) ,( 2,8,10) ,( 0,2,4)\} \end{array}
Then I ...
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Why Do Additive Categories Need Zero Objects? - Motivation
At the moment, I'm trying to develop intuition behind the derived construction of what is an additive category from the (standard) category definition.
(i) It seems natural that a category with the ...
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Finite Direct Sum in relation to Finite Direct Product - Additive Category
The question is have is admittedly very basic, but I struggle to find posts (on StackExchange) or proofs addressing it.
Let $A$ denote some arbitrary additive category. Hence, the direct sum $(X_1\...
- 917
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Triangle inequality on a direct sum of Lebesgue spaces
I think a priori it is known that for $1 \leq p, q < \infty$ we have that $||(\hspace{0.1cm},)||: L^p(E) \times L^q(E) \rightarrow \mathbb{R}_{\geq 0} $, defined as $||(f, g)|| = (||f||^2_{L^p} + |...
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