Is a linear operator on $\ell^2$ defined by the inner product necessarily bounded?
11986
If $\sum a_n b_n <\infty$ for all $(b_n)\in \ell^2$ then $(a_n) \in \ell^2$
38584
$T\colon L^2[0,1] \to L^2[0,1]$ be a bounded linear map of Hilbert spaces such that if $f\in L^2[0,1]$ is continuous then so is $Tf$.
33485
norm of integral operator in $C([0,1])$
31919
What can we say about the inner product $\langle f(t),f'(t) \rangle$?
129020
Problem with the Definition of Inner Product Space.
149664
Norm is sup of inner products (proof).
156858
Kernel of powers of a normal operator
93923
No surjective bounded linear map from $\ell^2(\mathbf{N})$ to $\ell^1(\mathbf{N})$
108781
$A$ and $B$ are bounded linear operators from the normed linear space $X$ to itself. If $AB$ is invertible are $A$ and $B$ invertible?
99942
A normal, idempotent linear operator must be self-adjoint
132869
Show that any bounded linear functional on a normed linear space is continuous
140813
How do i prove that continuous linear transformation between normed spaces is bounded?
6202
$V$ be a vector space , $T:V \to V$ be a linear operator , then is $\ker (T) \cap R(T) \cong R(T)/R(T^2) $?
157455
Show that the norm of the multiplication operator $M_f$ on $L^2[0,1]$ is $\|f\|_\infty$
55067
Norm of self-adjoint operator
156855
separable Banach space with Banach-Mazur distances to $\ell_2^n$ bounded must be isomorphic to $\ell_2$?
152196
About inner products, norms and metrics