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These are called Bochner spaces, and yes, they are Banach spaces, assuming $B$ is separable (thanks Gerald).

At least one of the standard proofs that $L^p$ is complete goes through basically without change:

Let $f_n$ be Cauchy in this norm. Pass to a subsequence so that $|\|f_n - f_{n+1}\|| \le 4^{-n}$. By Chebyshev's inequality, we then have $P(\|f_n - f_{n+1}\| \ge 2^{-n}) \le 2^{-pn}$. Then the Borel-Cantelli lemma implies that for almost every $\omega \in \Omega$, we have $\|f_n(\omega) - f_{n+1}(\omega)\| \le 2^{-n}$ for all but finitely many $n$. In particular, for such $\omega$, the sequence $\{f_n(\omega)\}$ is Cauchy in $B$, so it converges to some $f(\omega) \in B$.

Now you have that $f$ is the a.e. limit of the $f_n$. Let $\epsilon > 0$. Since $f_n$ is Cauchy in $|\|\cdot\||$-norm, choose $N$ so large that $|\|f_n - f_m\|| < \epsilon$ for all $n,m > N$. Letting $m \to \infty$ and using Fatou's lemma on the integrals $\int_\Omega \|f_n - f_m\|\,dP$, conclude that $|\|f_n - f\|| < \epsilon$ as well. Thus the subsequence $f_n$ converges to $f$ in norm. Now use the Cauchy assumption one more time to see that the original sequence converges to $f$ as well.

I think that Evans's PDE book has some basic results about these spaces. There should be lots of other functional analysis texts that discuss them in more detail.

These are called Bochner spaces. Under mild assumptions (see Gerald's post), they are Banach spaces.

It is sufficient to assume that $B$ is separable, or that $L^p(\Omega, B)$ is defined to include only functions with almost every value in a separable subspace. Without some assumptions, it is possible that your $L^p(\Omega, B)$ is not even a vector space.

Given such assumptions, then at least one of the standard proofs that $L^p$ is complete goes through basically without change:

Let $f_n$ be Cauchy in this norm. Pass to a subsequence so that $|\|f_n - f_{n+1}\|| \le 4^{-n}$. By Chebyshev's inequality, we then have $P(\|f_n - f_{n+1}\| \ge 2^{-n}) \le 2^{-pn}$. Then the Borel-Cantelli lemma implies that for almost every $\omega \in \Omega$, we have $\|f_n(\omega) - f_{n+1}(\omega)\| \le 2^{-n}$ for all but finitely many $n$. In particular, for such $\omega$, the sequence $\{f_n(\omega)\}$ is Cauchy in $B$, so it converges to some $f(\omega) \in B$.

Now you have that $f$ is the a.e. limit of the $f_n$. Let $\epsilon > 0$. Since $f_n$ is Cauchy in $|\|\cdot\||$-norm, choose $N$ so large that $|\|f_n - f_m\|| < \epsilon$ for all $n,m > N$. Letting $m \to \infty$ and using Fatou's lemma on the integrals $\int_\Omega \|f_n - f_m\|\,dP$, conclude that $|\|f_n - f\|| < \epsilon$ as well. Thus the subsequence $f_n$ converges to $f$ in norm. Now use the Cauchy assumption one more time to see that the original sequence converges to $f$ as well.

I think that Evans's PDE book has some basic results about these spaces. There should be lots of other functional analysis texts that discuss them in more detail.

These are called Bochner spaces, and yes, they are Banach spaces.

At least one of the standard proofs that $L^p$ is complete goes through basically without change:

Let $f_n$ be Cauchy in this norm. Pass to a subsequence so that $|\|f_n - f_{n+1}\|| \le 4^{-n}$. By Chebyshev's inequality, we then have $P(\|f_n - f_{n+1}\| \ge 2^{-n}) \le 2^{-pn}$. Then the Borel-Cantelli lemma implies that for almost every $\omega \in \Omega$, we have $\|f_n(\omega) - f_{n+1}(\omega)\| \le 2^{-n}$ for all but finitely many $n$. In particular, for such $\omega$, the sequence $\{f_n(\omega)\}$ is Cauchy in $B$, so it converges to some $f(\omega) \in B$.

Now you have that $f$ is the a.e. limit of the $f_n$. Let $\epsilon > 0$. Since $f_n$ is Cauchy in $|\|\cdot\||$-norm, choose $N$ so large that $|\|f_n - f_m\|| < \epsilon$ for all $n,m > N$. Letting $m \to \infty$ and using Fatou's lemma on the integrals $\int_\Omega \|f_n - f_m\|\,dP$, conclude that $|\|f_n - f\|| < \epsilon$ as well. Thus the subsequence $f_n$ converges to $f$ in norm. Now use the Cauchy assumption one more time to see that the original sequence converges to $f$ as well.

I think that Evans's PDE book has some basic results about these spaces. There should be lots of other functional analysis texts that discuss them in more detail.

These are called Bochner spaces, and yes, they are Banach spaces, assuming $B$ is separable (thanks Gerald).

At least one of the standard proofs that $L^p$ is complete goes through basically without change:

Let $f_n$ be Cauchy in this norm. Pass to a subsequence so that $|\|f_n - f_{n+1}\|| \le 4^{-n}$. By Chebyshev's inequality, we then have $P(\|f_n - f_{n+1}\| \ge 2^{-n}) \le 2^{-pn}$. Then the Borel-Cantelli lemma implies that for almost every $\omega \in \Omega$, we have $\|f_n(\omega) - f_{n+1}(\omega)\| \le 2^{-n}$ for all but finitely many $n$. In particular, for such $\omega$, the sequence $\{f_n(\omega)\}$ is Cauchy in $B$, so it converges to some $f(\omega) \in B$.

Now you have that $f$ is the a.e. limit of the $f_n$. Let $\epsilon > 0$. Since $f_n$ is Cauchy in $|\|\cdot\||$-norm, choose $N$ so large that $|\|f_n - f_m\|| < \epsilon$ for all $n,m > N$. Letting $m \to \infty$ and using Fatou's lemma on the integrals $\int_\Omega \|f_n - f_m\|\,dP$, conclude that $|\|f_n - f\|| < \epsilon$ as well. Thus the subsequence $f_n$ converges to $f$ in norm. Now use the Cauchy assumption one more time to see that the original sequence converges to $f$ as well.

I think that Evans's PDE book has some basic results about these spaces. There should be lots of other functional analysis texts that discuss them in more detail.

These are called Bochner spaces, and yes, they are Banach spaces. The proof that it's a Banach space should be pretty much identical to the proof for real-valued functions.

I think that Evans's PDE book has some basic results about these spaces. There should be lots of other functional analysis texts that discuss them in more detail.

These are called Bochner spaces, and yes, they are Banach spaces.

At least one of the standard proofs that $L^p$ is complete goes through basically without change:

Let $f_n$ be Cauchy in this norm. Pass to a subsequence so that $|\|f_n - f_{n+1}\|| \le 4^{-n}$. By Chebyshev's inequality, we then have $P(\|f_n - f_{n+1}\| \ge 2^{-n}) \le 2^{-pn}$. Then the Borel-Cantelli lemma implies that for almost every $\omega \in \Omega$, we have $\|f_n(\omega) - f_{n+1}(\omega)\| \le 2^{-n}$ for all but finitely many $n$. In particular, for such $\omega$, the sequence $\{f_n(\omega)\}$ is Cauchy in $B$, so it converges to some $f(\omega) \in B$.

Now you have that $f$ is the a.e. limit of the $f_n$. Let $\epsilon > 0$. Since $f_n$ is Cauchy in $|\|\cdot\||$-norm, choose $N$ so large that $|\|f_n - f_m\|| < \epsilon$ for all $n,m > N$. Letting $m \to \infty$ and using Fatou's lemma on the integrals $\int_\Omega \|f_n - f_m\|\,dP$, conclude that $|\|f_n - f\|| < \epsilon$ as well. Thus the subsequence $f_n$ converges to $f$ in norm. Now use the Cauchy assumption one more time to see that the original sequence converges to $f$ as well.

I think that Evans's PDE book has some basic results about these spaces. There should be lots of other functional analysis texts that discuss them in more detail.