Singular Value Decomposition - what do I have to do?

Question

Show that the Singular Value Decomposition of

$$ T\colon L^2([0,1])\to H^1([0,1]), x\mapsto\int\limits_0^t x(s)\, ds $$

is given by

$$ \sigma_j=\frac{1}{(j-1/2)\pi}, v_j(x)=\sqrt{2}\cos((j-1/2)\pi x), u_j(x)=\sqrt{2}\sin((j-1/2)\pi x). $$

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I do not know what i have to do here. Do I "only" have to calculate that

$Tu_j=\sigma_j v_j$

and that

$T^* v_j=\sigma_j u_j$ ?

Answer 1

Finding an SVD means having complete sets of left- and right- singular vectors. So you should also check that $(u_j)$ is a basis of $L^2$ and $(v_j)$ is a basis of $H^1$.