Articles of vector space isomorphism

Show that $T\to T^*$ is an isomorphism (Where T is a linear transform)

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 […]

Is $\mathbb{C}^2$ isomorphic to $\mathbb{R}^4$?

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] […]

Equivalent definitions of isometry

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 $ U \oplus W_1 = U \oplus W_2 \implies W_1 = W_2 $?

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?

$\hom(V,W)$ is canonic isomorph to $\hom(W^*, V^*)$

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 […]

An example of a Banach space isomorphic but not isometric to a dual Banach space

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. […]

Are two Hilbert spaces with the same algebraic dimension (their Hamel bases have the same cardinality) isomorphic?

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 […]

$V$ 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) $?

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 […]