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Let $F: \mathbb R^n \times \mathbb R^{m^2} \to \mathbb R^{m^2}$, $F_{kl}(x,y)=\sum_{ij} y_{ik}y_{jl}\beta_{ij}(x,y)+\alpha_{kl}(x)+y_{kl}$. Prove that for smooth function $\alpha_{kl}(x),\beta_{kl}(x,y)$, where $\alpha_{kl}(x)$ is vanishing in zero, function $F$ is smooth.

I wonder if it's not enough say that since $\alpha_{kl}(x),\beta_{kl}(x,y)$ are smooth then also $F_{kl}$ is smooth but I'm afraid that because of the presence $y_{kl}$ this should be proved more accurately.

William8649
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    The projections are smooth – dmtri Dec 16 '19 at 10:32
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    Remember that the sum of smooth functions is a smooth function and the product of smooth function is a smooth function. This is everything that you need. The map $y\mapsto y_{ij}$ is smooth. – Matheus Manzatto Dec 16 '19 at 10:33
  • if the sum is an infinite one, $F$ need not be smooth, e.g. Weierstrass function. The presence of the constant term won't affect smoothness – fGDu94 Dec 16 '19 at 10:59

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