I think I solved it, but I used a dirty trick, I’d like someone to review it, that would be great. Let $X,Y$ be linear spaces over field $F$. and $T:X \to Y$ a linear transformation.For each $T$ we define $T^*:Y^* \to X^*$ such that $T^*(\phi)(x)=\phi(Tx)$ where $x \in X, \phi \in Y^*$ Show that […]

Are $\mathbb{C}^2$ and $\mathbb{R}^4$ isomorphic to one another? Two vector spaces are isomorphic if and only if there exists a bijection between the two. We can define the linear map $T: \mathbb{C}^2 \mapsto \mathbb{R}^4$ as $$ T\left(\left[\begin{array}{cc} a + bi \\ c + di \end{array}\right]\right) = \left[\begin{array}{cccc} a \\ b \\ c \\ d \end{array}\right] […]

Consider a map $T:\mathbb{R}^2\to\mathbb{R}^2$ such that $\lVert T(x)\rVert=\lVert x\rVert$. Is this equivalent to stating that $\langle x, y\rangle=\langle T(x), T(y)\rangle$ for all $x,y\in\mathbb{R}^2$, where $\langle\cdot,\cdot\rangle$ denotes the usual inner product?

Is it true that if $ U \oplus W_1 = U \oplus W_2 $, then $ W_1 = W_2 $? I think that if $ U \oplus W_1 = U \oplus W_2 $, then u+w1=u+w2, so W1=W2. But did I make any mistakes?

Introduction My Semester just started and we have a new Professor for Linear Algebra II (replacing our former Professor). Apparently we are behind our schedule and thus we only had a brief introduction (5 Minutes on the Blackboard) about the Dual Space and canonical isomorphisms. In the current problem set there is this optional (not […]

I am wondering the following question: Let $X$ be a separable Banach space which is linearly isomorphic to a dual Banach space $Y^*$. Is there a Banach space $Z$ such that $X$ is lineraly isometric to the dual of $Z$: $X=Z^*$. I think that the answer is no, but I do not have a counterexample. […]

We know that two Hilbert spaces that have orthonormal bases of the same cardinality are isomorphic (as an inner product spaces). My question is: what can we say when we know that their Hamel bases have the same cardinality? It clearly implies they are isomorphic as vector spaces (just send a basis to a basis […]

Let $V$ be a vector space , $T:V \to V$ be a linear operator , then is it true that $\ker (T) \cap R(T) \cong R(T)/R(T^2) $ ( where $R(T)$ denotes the range of $T$ ) ? I know that the statement is true when $V$ is finite dimensional as I can show that if […]

Intereting Posts

Examples of fields of characteristic 0?
Proving a necessary and sufficient condition for compactness of a subset of $\ell^p$
Irreducibility of $x^p-x-c$
Accessible Intro to Random Matrix Theory (RMT)
(Possible) application of Sarason interpolation theorem
Modal set-theory
Primary ideals confusion with definition
Prove the opposite angles of a quadrilateral are supplementary implies it is cyclic.
Evaluating and proving $\lim_{x\to\infty}\frac{\sin x}x$
Intuition behind the Axiom of Choice
Proof of lack of pure prime producing polynomials.
What is the image near the essential singularity of z sin(1/z)?
Showing that the norm of the canonical projection $X\to X/M$ is $1$
Cancellation Law for External direct product
Uniform convergence of $x^n$ on $(-1,1)$.