Question

How can we show that $E\:\hat⊗_π F→E\:\hat⊗_ε\:F↪\mathfrak B(E',F')$ is injective iff $E\:\hat⊗_π F→(E'⊗_εF')'↪\mathfrak B(E',F')$ is injective?

score 155 · Answer 1 · 2026-04-29 12:16:22

How can we show that $E\:\hat⊗_π F→E\:\hat⊗_ε\:F↪\mathfrak B(E',F')$ is injective iff $E\:\hat⊗_π F→(E'⊗_εF')'↪\mathfrak B(E',F')$ is injective?

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1.4k

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score 142 · Answer 8 · 2026-04-29 12:15:16

If $L∈\mathfrak L(X⊗_πY,E)$, then $\sup_{\left\|x\right\|_X,\:\left\|y\right\|_Y\:≤\:1}\left\|L(x⊗y)\right\|_E=\sup_{π(u)\:≤\:1}\left\|Lu\right\|_E$

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score 539 · Answer 13 · 2026-05-10 08:15:55

539

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score 141 · Answer 14 · 2026-04-17 02:05:04

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score 810 · Answer 15 · 2026-05-01 00:41:00

810

Views

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Published on 01 May 2026 - 0:41

List Question

How can we show that $E\:\hat⊗_π F→E\:\hat⊗_ε\:F↪\mathfrak B(E',F')$ is injective iff $E\:\hat⊗_π F→(E'⊗_εF')'↪\mathfrak B(E',F')$ is injective?

Is the total variation norm of a measure equal to its norm as a bounded functional?

Help me understand this proof of Implicit Function Theorem on Banach spaces

The diagonal extraction procedure?

Is it a equivalent semi norm in Campanato space?

X^* closed under pointwise convergence.

To prove x_n is summable under the given norm

If $L∈\mathfrak L(X⊗_πY,E)$, then $\sup_{\left\|x\right\|_X,\:\left\|y\right\|_Y\:≤\:1}\left\|L(x⊗y)\right\|_E=\sup_{π(u)\:≤\:1}\left\|Lu\right\|_E$

Representation of tensors in the projective tensor product

Definition of the approximation property of a Banach space

Banach algebra of functions under composition

Convergence radius $\beta$=sup{|$\lambda$|: $\lambda$ $\in$ $\Bbb K$ , $(\sum_{k=0}^n c_k \lambda^k)_n$ bounded )

Can a linear subspace in Banach space be the union of several other subspaces?

$T:X\rightarrow X$ bounded linear operator ($X$ Banach space), $||T||\leq 1$ and $T(x_0)=x_0$, then $x_0\notin \{x-Tx \mid x\in X \}$

Finding spectrum of given operator

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