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$$C_7 : C_3$$ is a semi-direct product. $$C_p$$ is as described. A prime, congruent to 1 mod 3.

After finding the number of elements for this group, how do I extend the argument to $$p,q\equiv1\left(mod\ 3\right)$$, where $$G=(C_p:C_3\ )\times(C_q:C_3\ )$$Any help appreciated.

Let $X$ be a *-algebra with identity $e$, and let $e\in{X}$, $\lambda\in\mathbb{C}$. Can somebody show me how $\sigma(\lambda{e}-x)=\lambda-\sigma(x)$, where $\sigma(x)$ is the spectrum of an element. Thanks in advance.

Thanks for this. Although I an now a little confused as to why we wanted to find the numbers for $f(M_1)$ and so on. How does knowing this enable us to find the scalar homomorphism $f$?

Thanks. That now makes more sense. Can I ask how you got the two scalar homomorphisms that map to $a$ and $d$ respectively? I need to use these to show that the radical of $M$ is $\begin{bmatrix}0&b\\0&0\end{bmatrix}$, where $b\in\mathbb{C}$.

But my bases (which are the ones requested in the question) are $M_1=\begin{bmatrix} 1&0 \\ 0&1 \end{bmatrix}$ $M_2=\begin{bmatrix} 0&1 \\ 0&0 \end{bmatrix}$ $M_3=\begin{bmatrix} 0&0 \\ 0&1 \end{bmatrix}$

Thanks for this Opalg. I can't see how $M_1+M_3=I$ for the basis matrices I have determined. The question I am tackling, is from Maddox and he specifically requests that this basis be found and then used to find the set of all scalar homomorphisms of the set M.

Let M be the set of 2x2 matrices defined by M = {a b 0 d} where a, b and d are complex. I've found a basis for M but need to know how to find the set of scalar homomorphisms of M from these. I have the basis as M_1 = {1 0 0 1} M_2 = {0 1 0 0} and M_3 = {0 0...

Aha! I never thought to use Hahn Banach. Thanks

Thank you for replying to my question. I had considered using the Closed Graph theorem but could only manage to show that (T*f)(x)=f(y). I'm not sure if you can even proceed from here to show that Tx=y and therefore deduce that T is continuous. Is it even possible?

I'm not that familiar with the Uniform Boundedness Principle, so wouldn't know where or how to apply this result to the question I'm afraid.

The question, stated in full is thus: Let X be a complex Banach space and T in L(X,X), a linear operator not assumed continuous. You may take it without proof that (T*f)(x)=f(Tx), where x in X and f in X* (1) defines a linear operator T* in L(X*,X*). Note that, whilst the domain and...

Perhaps I should restate the final remark and say "how is this sufficient to show that T is continuous".

Let X be a complex Banach space and T in L(X,X) a linear operator. Assuming only that (T*f)(x)=f(Tx), where x in X and f in X* how can I prove that T is continuous?

Thanks girdav, but I'm still looking for a proof that this is the case.