In the category of Banach spaces, where the objects are Banach spaces and the morphisms are continuous linear maps, what are there coproducts? Are they the typical direct sum of Banach spaces? If so, does it accept finite and infinite coproducts?
2026-03-27 07:18:35.1774595915
What is the coproduct in the category of Banach spaces and continuous linear maps?
687 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in CATEGORY-THEORY
- (From Awodey)$\sf C \cong D$ be equivalent categories then $\sf C$ has binary products if and only if $\sf D$ does.
- Continuous functor for a Grothendieck topology
- Showing that initial object is also terminal in preadditive category
- Is $ X \to \mathrm{CH}^i (X) $ covariant or contravariant?
- What concept does a natural transformation between two functors between two monoids viewed as categories correspond to?
- Please explain Mac Lane notation on page 48
- How do you prove that category of representations of $G_m$ is equivalent to the category of finite dimensional graded vector spaces?
- Terminal object for Prin(X,G) (principal $G$-bundles)
- Show that a functor which preserves colimits has a right adjoint
- Show that a certain functor preserves colimits and finite limits by verifying it on the stalks of sheaves
Related Questions in BANACH-SPACES
- Problem 1.70 of Megginson's "An Introduction to Banach Space Theory"
- Is the cartesian product of two Hilbert spaces a Hilbert space?
- Why is $\lambda\mapsto(\lambda\textbf{1}-T)^{-1}$ analytic on $\rho(T)$?
- Is ${C}[0,1],\Bbb{R}$ homeomorphic to any $\Bbb{R^n}$, for an integer $n$?
- Identify $\operatorname{co}(\{e_n:n\in\mathbb N\})$ and $\overline{\operatorname{co}}(\{e_n : n\in\mathbb N\})$ in $c_0$ and $\ell^p$
- Theorem 1.7.9 of Megginson: Completeness is a three-space property.
- A weakly open subset of the unit ball of the Read's space $R$ (an infinite-dimensional Banach space) is unbounded.
- Separability of differentiable functions
- Showing $u_{\lambda}(x):= \left(\frac{\lambda}{{\lambda}^{2}+|x|^2}\right)^{\frac{n-2}{2}}$ is not sequentially compact in $L^{2^{*}}$
- Proving that a composition of bounded operator and trace class operator is trace class
Related Questions in DIRECT-SUM
- Finding subspaces with trivial intersection
- Direct sum and the inclusion property
- direct sum of injective hull of two modules is equal to the injective hull of direct sum of those modules
- What does a direct sum of tensor products look like?
- does the direct sum of constant sequences and null sequences gives convergent sequence Vector space
- Existence of Subspace so direct sum gives the orignal vector space.
- A matrix has $n$ independent eigenvectors $\Rightarrow\Bbb R^n$ is the direct sum of the eigenspaces
- $\dim(\mathbb{V}_1 \oplus ...\oplus \mathbb{V}_k) = \dim\mathbb{V}_1+...+\dim\mathbb{V}_k$
- Product/coproduct properties: If $N_1\simeq N_2$ in some category, then $N_1\times N_3\simeq N_2\times N_3$?
- Direct Sums of Abelian Groups/$R$-Modules
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
Yes, the coproduct of two Banach spaces $(X,\|~\|_X)$ and $(Y,\| ~ \|_Y)$ is $(X \oplus Y,\|~\|_{X\oplus Y})$ with $\|(x,y)\|_{X \oplus Y} = \|x\|_X + \|y\|_Y$. This is just an example for a suitable norm, equivalent norms such as $\sqrt{\|x\|_X^2 + \|y\|_Y^2}$ give isomorphic Banach spaces. The universal property is easily verified. It follows that finite coproducts exist. But infinite coproducts don't exist. I've proven this here recently for Hilbert spaces, cannot find the question right now (someone else?). The same argument applies here.