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Suppose $X$ is a Banach space not isomorphic to a Hilbert space. Can we always find a subspace of $X$ that is not isomorphic to a quotient of $X$?

Suppose $X$ is a separable infinite dimensional Banach space, $E$ a finite dimensional subspace which is $c$-complemented in $X$. Is the following true? For any $\varepsilon>0$, one can find $x\notin …

Does the parallelogram law for vectors of equal length imply the full parallelogram law? That is, if for all norm one vectors $x$ and $y$ in a Banach space $X$ it holds that $\lVert x-y\rVert^2+\lVert …

If $T$ is a bounded operator which is not strictly singular, acting on a separable Banach space $X$, can one always find an infinite dimensional, closed and complemented, subspace $Y$ such that $T$ re …

Suppose that $X$ and $Y$ are two Banach spaces, $S_{X}$ and $S_{Y}$ their unit spheres, and $f$ an onto isometry between $S_X$ and $S_Y$. Does it follow that $X$ and $Y$ are isometric?

Suppose a (separable) Banach space $X$ has the following property: If $P:X\to X$ is a bounded projection different from $I$ such that $\|P\|=1$, then $\|I-P\|=1$. Does this imply that $X$ is a Hilbert …

I am not sure if this question makes sense, or if it is trivial, but does there exists an infinite dimensional Banach space (necessarily without the approximation property) such that no compact, non- …

If $X$ is a separable Banach space and $(x_n)$ is a basic sequence, then we can define biorthogonal functionals $(x^{*}_n)$ in $X^{*}$ such that $x^{*}_n(x_k)=\delta_{nk}$. What about conversely? If …

If $X$ is a separable Banach space and $(\epsilon_n)\downarrow 0$, can we find a quasinilpotent, non-compact operator on $X$ such that $||T^n||^{1/n}<\epsilon_n$ for all $n$? I suspect the answer is p …

Suppose $K$ is a centrally symmetric convex body in $\mathbb{R}^n$ and $E$ is the John's ellipsoid, the ellipsoid of maximal volume inside $K$. If $E$ and $K$ have exactly $2n$ contact points, say $(\ …

I asked this question on Math Stackexchange, but I didn't get an answer: https://math.stackexchange.com/questions/4881155/on-a-property-for-normed-spaces?noredirect=1#comment10410489_4881155 I came up …

This is a follow up to the question: Biorthogonal functionals. A positive answer to that question implies a negative answer to this one. If $X$ is a separable Banach space, can we find a basic seque …

If $X$ is an infinite dimensional (separable) Banach space, can one find a Hamel basis $(x_\alpha)_{\alpha\in\Lambda}$ such that all coordinate functionals $x_\alpha^*(x_\beta)=\delta_{\alpha,\beta}$ …

If $Y\subset X^*$ is a closed subspace (where $X$ is a separable Banach space), the preannihilator of $Y$ in $X$ is $Y_{\perp}:=\{x\in X : y^*(x)=0, \forall y^*\in Y \}$. If $Y$ is a proper subspace …

If $x_n$ is a normalized, weakly-null sequence in a Banach space, and $\epsilon_n\to 0$, does there exists a non-zero functional $f$ such that $|f(x_n)|<\epsilon_n$ for all $n$?