Let X be a Banach space, and x,y ∈ X. Show that if ℓ(x) = ℓ(y) for every ℓ ∈ X′, then x = y.
104607
Hahn-Banach theorem, dual space
59424
Showing that $B=\{(x,y) \in \Bbb{R^2} \mid \ xy>0 \}$ is an open set
90435
Prove that there exist linear functionals $L_1, L_2$ on $X$
36314
If $Ax = Bx$ for all $x \in C^{n}$, then $A = B$.
100161
Is it true that $\dim(X) \leq \dim(X^{\ast})$ for every infinite dimentional Banach space $X$?
61472
If $0<x<y$, then prove that $\sqrt{x} <\sqrt{y}$ and $x <\sqrt{xy} <y$
662
If $E[X|Y]=Y$ almost surely and $E[Y|X]=X$ almost surely then $X=Y$ almost surely
105316
Prove : $|x - y| \leqslant \epsilon$ $\forall \epsilon > 0$ iff $x = y$
34173
Banach spaces and their unit sphere
151046
Closure of the invertible operators on a Banach space
67283
Is $L^{p}$ space with alternate norm Banach?
106408
Prove that if W is a subspace of a finite dimensional vector space V, then dim(W) ≤ dim(V).
134859
Let $(X, \mathcal{T})$ be a metrizable space. Prove that $X$ Lindelöf implies that $X$ is separable.
109046
Is the complement of an open ball in a Banach space connected?
97949
Existence of a $\mathbb C$-Banach space isometric to a Hilbert Space but whose norm is not induced by an inner product?
74713
Show that for any natural numbers, if $S(x)=S(y)$ then $x=y$.
11942
show that $x$ is in $\ell^1$ if $\sum x_ny_n$ is convergent for all $y=(y_n)\in{c_0}$