The United States and the People's Republic of China established full diplomatic relations yesterday with low-key ceremonies and subdued references to the Chinese invasion of Vietnam.

Ceremonies in Washington and Peking symbolized the conversion of the liaison offices of the two countries to full-fledged embassies.

"I consider this to be a momentous day in the historical evolution of our nation," President Carter said when Chinese Ambassador Chai Tse-min presented his credentials at the White House. "We consider this to be a great opportunity for the future."

In a formal, written exchange that was not orally delivered but was made public later, Carter stressed the value of diplomacy because "the threat of war intrudes constantly on our daily lives." This statement was repeated in a toast last night by Secretary of Commerce Juanita M. Kreps, the ranking U.S. offical a Chinese reception at the embassy on Connecticut Avenue.

After the presentation of credentials in the Oval Office, Carter and Chai held a private talk in which the Chinese military action in Vietnam was discussed, accordin to White House sources. Officials said the topic also was raised in a meeting between Vice President Mondale and Tang Ke, the visiting Chinese minister of metallurgical industry.

Chai's public comments were unusually restrained, both at the White restrained, both at the White House and the Chinese reception. There was no mention of hegemony, the Chinese code word for Soviet domination, which has been included in formal communiques between Peking and Washington, and which was a central theme of the U.S. trip a month ago of Vice Premier Teng Hsiao-ping.

At the embassy reception, Chai said his government will protect the "legitimate rights and interests" of the overseas Chinese and requests them to work for "the return of Taiwan to the embrace of the motherland."

The ambassador, in a bid for most-favored-nation benefits under U.S. trade laws, pledged the Peking government to adopt "positive and effective meaasures" to reunite families and permit family visits from overseas Chinese. Under the Jackson-Vanik amendment to trade laws, benefits are tied to immigration policies in the case of communist countries.

The tone of the observance, and of Sino-American relations, was less festive than when agreement on the normalization of relations was announced in mid-December and when Teng vistited.

China's decision to invade Vietnam introduced a negative element, and brought public and private expressions of concern and disagreement from the United States. The military action has generated worry that a widening war in Asia could bring in the Soviet Union and cause worldwide crisis.

The United States' relations with Taiwan also underwant an official change in the past two days, with the end of formal diplomatic ties and closing of the U.S. embassy in Taipei.

The two countries will continue relations on an unofficial basis through private agencies set up in Washington and Taipei.

The scheduled opening yesterday of the U.S. agency, known as the American Institute on Taiwan, had to be postponed because Congress has yet to approve funding for it.

Final congressional action is expected to take a week or longer. State Department spokesman Hodding Carter said that in the interim period U.S. dealings with Taiwan will be carried out through the Taiwanese agency, designated the Coordination Council for North American Affairs.

The two entities will oversee all trade, cultural and financial relations between Taiwan and the United States. For the most part, both will employ government officials who will be granted leaves of absence during their service at the respective agencies.