# Questions tagged [direct-sum]

For questions about taking the direct sum of groups and other algebraic structures.

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### Why is it difficult to define a direct integral of Banach spaces or Banach algebras?

In https://en.wikipedia.org/wiki/Direct_integral I can read about how to define a direct integral on Hilbert spaces and Von-Neumann algebras. Suppose that I want to define a direct integral on either ...
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### Need help for understanding a sum of subspaces

I am newbie started learning the linear algebra. It might be dumb question. But I don't understand how the sum of subspace can also be subspace?! So for subset in order to be subspace, It should ...
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### Simple question- Diagonalization and direct sum of image and kernel

Is the following statement somewhat true? A matrix $A\in M_n(\mathbb{F})$ is diagnozable if (or iff) $Ker(A)\bigoplus Im(A)=\mathbb{R}^n$ or $=M_n$ not quite sure on what exactly the result of the ...
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### Direct sum of two elements of subspaces of a vector space

Help me to understand what the authors of this paper (p. 3) mean by the direct sum of two elements in a vector space. Let $X$ be a vector space with subspaces $Y$ and $Z$ Definition: X is a direct sum ...
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1 vote
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### Proof verification - proving a matrix is diagonalizable using representation theory

Let $A\in M_n(\mathbb{C})$ be a matrix s.t $A^N=I_n$. Prove, using representation theory, that $A$ is diagonalizable. My attempt: We look at $G=\langle A\rangle\subset GL_n(\mathbb{C})$. This is a ...
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### Direct sum of reproducing kernel Hilbert spaces (RKHS)

I am currently diving into the theory of reproducing kernel Hilbert spaces and am just at the beginning of understanding the background of reproducing kernels. I have stumbled upon the following ...
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### Making sense of a derivative on direct sum

Consider $\mathbb{R}^n=E(t)\bigoplus F(t)$ be a continuous splitting. Is that possible to define a derivative on the direct sum such that $$(x(t)\bigoplus y(t))'=x'(t)\bigoplus y'(t)$$ The reason ...