Do we have cross-language vector space for word embedding? Do we have cross-language vector space for word embedding?

When measure similarity for apple/Pomme/mela/Lacus/苹果/りんご, they should be the same

If would be great if there's available internet service of neuron network which already be trained by multiple language
You can try to read about MUSE (Multilingual Unsupervised and Supervised Embeddings) by Facebook. You can read it from its Github or this article. They also provide the FastText dictionary format (.vec file) for some languages. 

Their original paper shows how it aligns the vector of words from two different languages: