# Questions tagged [bochner-spaces]

For question involving Bochner space, which are generalization $\mathbb L^p$ spaces in the sense that the values of the functions are themselves in function spaces.

140 questions
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### Weak star convergent sequence in $L^\infty(0,T; L^2(\Omega))$

Given a sequence $(u_n)_{n\in \mathbb{N}} \subseteq L^\infty (0,T; L^2(\Omega)) \cap H^1(0,T;L^2(\Omega))$ with \begin{align*} u_n \overset{\ast}{\rightharpoonup} u \,\, \text{ in } \,\, L^\infty (0,...
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### The restriction of a variable in a Sobolev function is a.e. Sobolev

Let $I,J$ be bounded open intervals in $\mathbb{R}$. How to show that $W^{1,1}(I\times J)$ is embedded in the Bochner space $L^1(I, W^{1,1}(J))$? Is there a standard reference where this is ...
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### Absolutely continuous and differentiable almost everywhere

I've read the following claim and I wonder if someone can direct me to or provide me with a proof of it: "A strongly absolutely continuous function which is differentiable almost everywhere is ...
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### Dual pairing Bochner spaces/ negative Sobolev spaces

Assuming $f \in L^2((0,T)\times \Omega)$ and $g \in L^2(0,T;W^{1,2}(\Omega))$, is the following correct? \begin{equation} \int_0^T\int_\Omega fgdxdt = <f,g>_{L^2(0,T;W^{-1,2}(\Omega)) \times L^...
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### Integration in normed vector spaces

Given an interval $I\subset \mathbb{R}$ and a normed vector space $X$, I want to know if I am able to define the Lebesgue space $L^p(I;X)$ of all $p$-integrable functions $f:I\to X$. I know that this ...
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### How to prove that, $T\left(\int_a^b f(t)dt\right) =\int_a^b T\circ f(t)dt$

Let $T:E\to F$ be a linear and continuous function between the Banach spaces $E$ and $F$. We consider $f:\to [a,b]\to E$ (with $a<b$) be an integrable function in the sense of Bochner How do I ...
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### Product rule and integration by parts in Bochner Sobolev space $W^{1,1}(U)$

Assume $v(t)\in H_0^1(U)$ for all $0\leq t\leq T$. The weak derivative $v'$ of $v$ is defined by \begin{equation} \int_0^T \phi'(t)v(t)\,\mathrm{d}t=\int_0^T\phi(t)v'(t)\,\mathrm{d}t\quad\text{for all ...
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### Maximal monotone operators - Zeidler's book - question about the proof.

I have a problem with understanding the proof of the proposition 32.10 which I enclose below (E. Zeidler II B). I cannot understand why Zeidler gets from (28) \int_{0}^{T}\varphi'(t)v(t)+\varphi(t)...
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### Existence of time derivative in Bochner space $L^2(0,T;H_0^1(U))$ (parabolic pde)

Evans defines a function $u=[u(t)](x):=u(t,x)$, \begin{equation} u\in L^2(0,T;H_0^1(U))\quad\text{with}\quad u'\in L^2(0,T;H^{-1}(U)) \end{equation} as a weak solution of a certain parabolic initial/...
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### Dual of an operator

A linear operator $A$ is defined from $L^2(0,T)$ to $C([0,T);\mathbb{H})$ where $\mathbb{H}$ is a Hilbert space. At some point, dual of this operator is needed; a detour to calculate the dual, and ...
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### Can someone help me obtain these a-priori estimates?

I am trying to obtain some estimates on a PDE where the unknown is: $u(t) : \mathbb{R}^n \rightarrow \mathbb{R}^n, t \in [0,T]$. The variational formulation is given below, where the test function ...
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### If a process $Y=\{Y_{t}:t\in[0,T]\}$ is progressive, then $\{\int_{0}^{t}Y(s)\text{ d}s\}$ is *surely* continuous

I am reading a textbook and I am stuck proving a seemingly trivial point (but I believe I am not the only one that was pondering this question and it shows similarity to this question). Let me sketch ...
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### About the dual space of $V=\{u\in H_0^1(\Omega): \text{div}u=0 \}$ and its relations to $H^{-1}(\Omega)$.

I read about some things about $V=\{u\in H_0^1(\Omega): \text{div}u=0 \}$ and its dual space and I began to mix some of these things together. As a result: irritation. I hope you can help me out. ...
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### Reference for Bochner space.

Are there any books that has a nice introduction to Bochner space including its properties and proofs? Not Evans PDEs. One of my friend recommended me this: https://books.google.co.uk/books?id=...
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### Is $L^1(0,T;V') \subset H^{-1}(0,T;H)$ for $V \subset H \subset V'$?

My question is about a passage in the paper 'On the existence of the pressure for solutions of the variational Navier–Stokes equations' by J.Simon. (It can be found easily via Google). It says the ...